We cannot always calculate antiderivative functions, but the differentiation problem can be solved for any function. That is why there is no single integration method that can be used for any type of calculation.
In this material, we will look at examples of solving problems related to finding the indefinite integral, and see what types of integrands each method is suitable for.
Direct integration method
The main method for calculating the antiderivative function is direct integration. This action is based on the properties of the indefinite integral, and for the calculations we need a table of antiderivatives. Other methods can only help bring the original integral to tabular form.
Example 1
Calculate the set of antiderivatives of the function f (x) = 2 x + 3 2 · 5 x + 4 3 .
Solution
First, let's change the form of the function to f (x) = 2 x + 3 2 5 x + 4 3 = 2 x + 3 2 5 x + 4 1 3.
We know that the integral of the sum of functions will be equal to the sum of these integrals, which means:
∫ f (x) d x = ∫ 3 2 5 x + 4 3 = 2 x + 3 2 5 x + 4 1 3 d x = ∫ 3 2 5 x + 4 1 3 d x
We derive the numerical coefficient behind the integral sign:
∫ f (x) d x = ∫ 2 x d x + ∫ 3 2 (5 x + 4) 1 3 d x = = ∫ 2 x d x + 2 3 ∫ (5 x + 4) 1 3 d x
To find the first integral, we will need to refer to the table of antiderivatives. We take from it the value ∫ 2 x d x = 2 x ln 2 + C 1
To find the second integral, you will need a table of antiderivatives for the power function ∫ x p · d x = x p + 1 p + 1 + C , as well as the rule ∫ f k · x + b d x = 1 k · F (k · x + b) + C .
Therefore, ∫ f (x) d x = ∫ 2 x d x + 3 2 ∫ 5 x + 4 1 3 d x = = 2 x ln 2 + C 1 + 3 2 3 20 (5 x + 4) 4 3 + C 2 = = 2 x ln 2 + 9 40 5 x + 4 4 3 + C
We got the following:
∫ f (x) d x = ∫ 2 x d x + 3 2 ∫ 5 x + 4 1 3 d x = = 2 x ln 2 + C 1 + 3 2 3 20 (5 x + 4) 4 3 + C 2 = = 2 x ln 2 + 9 40 5 x + 4 4 3 + C
with C = C 1 + 3 2 C 2
Answer:∫ f (x) d x = 2 x ln 2 + 9 40 5 x + 4 4 3 + C
We devoted a separate article to direct integration using tables of antiderivatives. We recommend that you familiarize yourself with it.
Substitution method
This method of integration consists in expressing the integrand through a new variable introduced specifically for this purpose. As a result, we should get a tabular form of the integral or simply a less complex integral.
This method is very useful when you need to integrate functions with radicals or trigonometric functions.
Example 2
Evaluate the indefinite integral ∫ 1 x 2 x - 9 d x .
Solution
Let's add one more variable z = 2 x - 9 . Now we need to express x in terms of z:
z 2 = 2 x - 9 ⇒ x = z 2 + 9 2 ⇒ d x = d z 2 + 9 2 = z 2 + 9 2 " d z = 1 2 z d z = z d z
∫ d x x 2 x - 9 = ∫ z d z z 2 + 9 2 · z = 2 ∫ d z z 2 + 9
We take the table of antiderivatives and find out that 2 ∫ d z z 2 + 9 = 2 3 a r c t g z 3 + C .
Now we need to return to the variable x and get the answer:
2 3 a r c t g z 3 + C = 2 3 a r c t g 2 x - 9 3 + C
Answer:∫ 1 x 2 x - 9 d x = 2 3 a r c t g 2 x - 9 3 + C .
If we have to integrate functions with irrationality of the form x m (a + b x n) p, where the values m, n, p are rational numbers, then it is important to correctly formulate an expression for introducing a new variable. Read more about this in the article on integrating irrational functions.
As we said above, the substitution method is convenient to use when you need to integrate a trigonometric function. For example, using a universal substitution, you can reduce an expression to a fractionally rational form.
This method explains the integration rule ∫ f (k · x + b) d x = 1 k · F (k · x + b) + C .
We add another variable z = k x + b. We get the following:
x = z k - b k ⇒ d x = d z k - b k = z k - b k " d z = d z k
Now we take the resulting expressions and add them to the integral specified in the condition:
∫ f (k x + b) d x = ∫ f (z) d z k = 1 k ∫ f (z) d z = = 1 k F z + C 1 = F (z) k + C 1 k
If we accept C 1 k = C and return to the original variable x, then we get:
F (z) k + C 1 k = 1 k F k x + b + C
Method of subscribing to the differential sign
This method is based on transforming the integrand into a function of the form f (g (x)) d (g (x)). After this, we perform a substitution by introducing a new variable z = g (x), find its antiderivative and return to the original variable.
∫ f (g (x)) d (g (x)) = g (x) = z = ∫ f (z) d (z) = = F (z) + C = z = g (x) = F ( g(x)) + C
To solve problems faster using this method, keep a table of derivatives in the form of differentials and a table of antiderivatives on hand to find the expression to which the integrand will need to be reduced.
Let us analyze a problem in which we need to calculate the set of antiderivatives of the cotangent function.
Example 3
Calculate the indefinite integral ∫ c t g x d x .
Solution
Let's transform the original expression under the integral using basic trigonometric formulas.
c t g x d x = cos s d x sin x
We look at the table of derivatives and see that the numerator can be subsumed under the differential sign cos x d x = d (sin x), which means:
c t g x d x = cos x d x sin x = d sin x sin x, i.e. ∫ c t g x d x = ∫ d sin x sin x .
Let us assume that sin x = z, in this case ∫ d sin x sin x = ∫ d z z. According to the table of antiderivatives, ∫ d z z = ln z + C . Now let's return to the original variable ∫ d z z = ln z + C = ln sin x + C .
The entire solution can be briefly written as follows:
∫ с t g x d x = ∫ cos x d x sin x = ∫ d sin x sin x = s i n x = t = = ∫ d t t = ln t + C = t = sin x = ln sin x + C
Answer: ∫ c t g x d x = ln sin x + C
The method of subscribing to the differential sign is very often used in practice, so we advise you to read a separate article dedicated to it.
Method of integration by parts
This method is based on transforming the integrand into a product of the form f (x) d x = u (x) v " x d x = u (x) d (v (x)), after which the formula ∫ u (x) d ( v (x)) = u (x) · v (x) - ∫ v (x) · d u (x).This is a very convenient and common solution method. Sometimes partial integration in one problem has to be applied several times before obtaining the desired result.
Let us analyze a problem in which we need to calculate the set of antiderivatives of the arctangent.
Example 4
Calculate the indefinite integral ∫ a r c t g (2 x) d x .
Solution
Let's assume that u (x) = a r c t g (2 x), d (v (x)) = d x, in this case:
d (u (x)) = u " (x) d x = a r c t g (2 x) " d x = 2 d x 1 + 4 x 2 v (x) = ∫ d (v (x)) = ∫ d x = x
When we calculate the value of the function v (x), we should not add an arbitrary constant C.
∫ a r c t g (2 x) d x = u (x) v (x) - ∫ v (x) d (u (x)) = = x a r c t g (2 x) - ∫ 2 x d x 1 + 4 x 2
We calculate the resulting integral using the method of subsuming the differential sign.
Since ∫ a r c t g (2 x) d x = u (x) · v (x) - ∫ v (x) d (u (x)) = x · a r c t g (2 x) - ∫ 2 x d x 1 + 4 x 2 , then 2 x d x = 1 4 d (1 + 4 x 2) .
∫ a r c t g (2 x) d x = x · a r c t g (2 x) - ∫ 2 x d x 1 + 4 x 2 = = x · a r c t g (2 x) - 1 4 ln 1 + 4 x 2 + C 1 = = x · a r c t g (2 x) - 1 4 ln 1 + 4 x 2 + C
Answer:∫ a r c t g (2 x) d x = x · a r c t g (2 x) - 1 4 ln 1 + 4 x 2 + C .
The main difficulty in using this method is the need to choose which part to take as the differential and which part as the function u (x). The article on the method of integration by parts provides some advice on this issue that you should familiarize yourself with.
If we need to find the set of antiderivatives of a fractionally rational function, then we must first represent the integrand as a sum of simple fractions, and then integrate the resulting fractions. For more information, see the article on integrating simple fractions.
If we integrate a power expression of the form sin 7 x · d x or d x (x 2 + a 2) 8, then we will benefit from recurrence formulas that can gradually lower the power. They are derived using sequential repeated integration by parts. We recommend reading the article “Integration using recurrence formulas.
Let's summarize. To solve problems, it is very important to know the method of direct integration. Other methods (substitution, substitution, integration by parts) also allow you to simplify the integral and bring it to tabular form.
If you notice an error in the text, please highlight it and press Ctrl+Enter
In this topic we will talk in detail about the properties of the indefinite integral and about finding the integrals themselves using the mentioned properties. We will also work with the table of indefinite integrals. The material presented here is a continuation of the topic "Indefinite integral. Beginning". To be honest, test papers rarely contain integrals that can be taken using typical tables and/or simple properties. These properties can be compared to the alphabet, knowledge and understanding of which is necessary to understand the mechanism for solving integrals in other topics. Often integration using tables of integrals and properties of the indefinite integral is called direct integration.
What I'm getting at: the functions change, but the formula for finding the derivative remains unchanged, unlike the integral, for which we already had to list two methods.
Let's go further. To find the derivative $y=x^(-\frac(1)(2))\cdot(1+x^(\frac(1)(4)))^\frac(1)(3)$ all the same applies the same formula $(u\cdot v)"=u"\cdot v+u\cdot v"$, into which you will have to substitute $u=x^(-\frac(1)(2))$, $v=( 1+x^(\frac(1)(4)))^\frac(1)(3)$. But to find the integral $\int x^(-\frac(1)(2))\cdot( 1+x^(\frac(1)(4)))^\frac(1)(3) dx$ will require the use of a new method - Chebyshev substitutions.
And finally: to find the derivative of the function $y=\sin x\cdot\frac(1)(x)$, the formula $(u\cdot v)"=u"\cdot v+u\cdot v"$ is again applicable, into which instead of $u$ and $v$ we substitute $\sin x$ and $\frac(1)(x)$, respectively. But $\int \sin x\cdot\frac(1)(x) dx$ is not is taken, or more precisely, is not expressed in terms of a finite number of elementary functions.
Let's summarize: where one formula was needed to find the derivative, four were required for the integral (and this is not the limit), and in the latter case the integral refused to be found at all. The function was changed - a new integration method was needed. This is where we have multi-page tables in reference books. The lack of a general method (suitable for solving “manually”) leads to an abundance of private methods that are applicable only for integrating their own, extremely limited class of functions (in further topics we will deal with these methods in detail). Although I cannot help but note the presence of the Risch algorithm (I advise you to read the description on Wikipedia), it is only suitable for program processing of indefinite integrals.
Question #3
But if there are so many of these properties, how can I learn to take integrals? It was easier with derivatives!
For a person, there is only one way so far: to solve as many examples as possible using various integration methods, so that when a new indefinite integral appears, you can choose a solution method for it based on your experience. I understand that the answer is not very reassuring, but there is no other way.
Properties of the indefinite integral
Property No. 1
The derivative of the indefinite integral is equal to the integrand, i.e. $\left(\int f(x) dx\right)"=f(x)$.
This property is quite natural, since the integral and derivative are mutually inverse operations. For example, $\left(\int \sin 3x dx\right)"=\sin 3x$, $\left(\int \left(3x^2+\frac(4)(\arccos x)\right) dx\ right)"=3x^2+\frac(4)(\arccos x)$ and so on.
Property No. 2
The indefinite integral of the differential of some function is equal to this function, i.e. $\int \mathrm d F(x) =F(x)+C$.
Usually this property is perceived as somewhat difficult, since it seems that there is “nothing” under the integral. To avoid this, you can write the indicated property as follows: $\int 1\mathrm d F(x) =F(x)+C$. An example of using this property: $\int \mathrm d(3x^2+e^x+4)=3x^2+e^x+4+C$ or, if you like, in this form: $\int 1\; \mathrm d(3x^2+e^x+4) =3x^2+e^x+4+C$.
Property No. 3
The constant factor can be taken out of the integral sign, i.e. $\int a\cdot f(x) dx=a\cdot\int f(x) dx$ (we assume that $a\neq 0$).
The property is quite simple and, perhaps, does not require comments. Examples: $\int 3x^5 dx=3\cdot \int x^5 dx$, $\int (2x+4e^(7x)) dx=2\cdot\int(x+2e^(7x))dx $, $\int kx^2dx=k\cdot\int x^2dx$ ($k\neq 0$).
Property No. 4
The integral of the sum (difference) of two functions is equal to the sum (difference) of the integrals of these functions:
$$\int(f_1(x)\pm f_2(x))dx=\int f_1(x)dx\pm\int f_2(x)dx$$
Examples: $\int(\cos x+x^2)dx=\int \cos xdx+\int x^2 dx$, $\int(e^x - \sin x)dx=\int e^xdx -\ int \sin x dx$.
In standard tests, properties No. 3 and No. 4 are usually used, so we will dwell on them in more detail.
Example No. 3
Find $\int 3 e^x dx$.
Let's use property No. 3 and take out the constant, i.e. number $3$, for the integral sign: $\int 3 e^x dx=3\cdot\int e^x dx$. Now let's open the table of integrals and substituting $u=x$ into formula No. 4 we get: $\int e^x dx=e^x+C$. It follows that $\int 3 e^x dx=3\cdot\int e^x dx=3e^x+C$. I assume that the reader will immediately have a question, so I will formulate this question separately:
Question #4
If $\int e^x dx=e^x+C$, then $\int 3 e^x dx=3\cdot\int e^x dx=3\cdot\left(e^x+C\right) =3e^x+3C$! Why did they just write $3e^x+C$ instead of $3e^x+3C$?
The question is completely reasonable. The point is that the integral constant (i.e. that same number $C$) can be represented in the form of any expression: the main thing is that this expression “runs through” the entire set of real numbers, i.e. varied from $-\infty$ to $+\infty$. For example, if $-\infty≤ C ≤ +\infty$, then $-\infty≤ \frac(C)(3) ≤ +\infty$, so the constant $C$ can be represented in the form $\frac(C)( 3)$. We can write that $\int e^x dx=e^x+\frac(C)(3)$ and then $\int 3 e^x dx=3\cdot\int e^x dx=3\cdot\left (e^x+\frac(C)(3)\right)=3e^x+C$. As you can see, there is no contradiction here, but you need to be careful when changing the form of the integral constant. For example, representing the constant $C$ as $C^2$ would be an error. The point is that $C^2 ≥ 0$, i.e. $C^2$ does not change from $-\infty$ to $+\infty$ and does not “run through” all real numbers. Likewise, it would be a mistake to represent a constant as $\sin C$, because $-1≤ \sin C ≤ 1$, i.e. $\sin C$ does not "run" through all values of the real axis. In what follows, we will not discuss this issue in detail, but will simply write the constant $C$ for each indefinite integral.
Example No. 4
Find $\int\left(4\sin x-\frac(17)(x^2+9)-8x^3 \right)dx$.
Let's use property No. 4:
$$\int\left(4\sin x-\frac(17)(x^2+9)-8x^3 \right) dx=\int 4\sin x dx-\int\frac(17)(x ^2+9)dx-\int8x^3dx$$
Now let’s take the constants (numbers) outside the integral signs:
$$\int 4\sin x dx-\int\frac(17)(x^2+9)dx-\int8x^3dx=4\int \sin x dx-17\int\frac(dx)(x^ 2+9)-8\int x^3dx$$
Next, we will work with each obtained integral separately. The first integral, i.e. $\int \sin x dx$, can be easily found in the table of integrals under No. 5. Substituting $u=x$ into formula No. 5 we get: $\int \sin x dx=-\cos x+C$.
To find the second integral $\int\frac(dx)(x^2+9)$ you need to apply formula No. 11 from the table of integrals. Substituting $u=x$ and $a=3$ into it we get: $\int\frac(dx)(x^2+9)=\frac(1)(3)\cdot \arctg\frac(x)( 3)+C$.
And finally, to find $\int x^3dx$ we use formula No. 1 from the table, substituting $u=x$ and $\alpha=3$ into it: $\int x^3dx=\frac(x^(3 +1))(3+1)+C=\frac(x^4)(4)+C$.
All integrals included in the expression $4\int \sin x dx-17\int\frac(dx)(x^2+9)-8\int x^3dx$ have been found. All that remains is to substitute them:
$$4\int \sin x dx-17\int\frac(dx)(x^2+9)-8\int x^3dx=4\cdot(-\cos x)-17\cdot\frac(1) (3)\cdot\arctg\frac(x)(3)-8\cdot\frac(x^4)(4)+C=\\ =-4\cdot\cos x-\frac(17)(3 )\cdot\arctg\frac(x)(3)-2\cdot x^4+C.$$
The problem is solved, the answer is: $\int\left(4\sin x-\frac(17)(x^2+9)-8x^3 \right)dx=-4\cdot\cos x-\frac(17 )(3)\cdot\arctg\frac(x)(3)-2\cdot x^4+C$. I will add one small note to this problem:
Just a small note
Perhaps no one will need this insert, but I’ll still mention that $\frac(1)(x^2+9)\cdot dx=\frac(dx)(x^2+9)$. Those. $\int\frac(17)(x^2+9)dx=17\cdot\int\frac(1)(x^2+9)dx=17\cdot\int\frac(dx)(x^2 +9)$.
Let's look at an example in which we use formula No. 1 from the table of integrals to interpose irrationalities (roots, in other words).
Example No. 5
Find $\int\left(5\cdot\sqrt(x^4)-\frac(14)(\sqrt(x^6))\right)dx$.
To begin with, we will do the same actions as in example No. 3, namely: we will decompose the integral into two and move the constants beyond the signs of the integrals:
$$\int\left(5\cdot\sqrt(x^4)-\frac(14)(\sqrt(x^6)) \right)dx=\int\left(5\cdot\sqrt(x^ 4) \right)dx-\int\frac(14)(\sqrt(x^6)) dx=\\ =5\cdot\int\sqrt(x^4) dx-14\cdot\int\frac( dx)(\sqrt(x^6)) $$
Since $\sqrt(x^4)=x^(\frac(4)(7))$, then $\int\sqrt(x^4) dx=\int x^(\frac(4)(7 ))dx$. To find this integral, we apply formula No. 1, substituting $u=x$ and $\alpha=\frac(4)(7)$ into it: $\int x^(\frac(4)(7))dx=\ frac(x^(\frac(4)(7)+1))(\frac(4)(7)+1)+C=\frac(x^(\frac(11)(7)))(\ frac(11)(7))+C=\frac(7\cdot\sqrt(x^(11)))(11)+C$. If you wish, you can represent $\sqrt(x^(11))$ as $x\cdot\sqrt(x^(4))$, but this is not necessary.
Let us now turn to the second integral, i.e. $\int\frac(dx)(\sqrt(x^6))$. Since $\frac(1)(\sqrt(x^6))=\frac(1)(x^(\frac(6)(11)))=x^(-\frac(6)(11) )$, then the integral under consideration can be represented in the following form: $\int\frac(dx)(\sqrt(x^6))=\int x^(-\frac(6)(11))dx$. To find the resulting integral, we apply formula No. 1 from the table of integrals, substituting $u=x$ and $\alpha=-\frac(6)(11)$ into it: $\int x^(-\frac(6)(11) ))dx=\frac(x^(-\frac(6)(11)+1))(-\frac(6)(11)+1)+C=\frac(x^(\frac(5) (11)))(\frac(5)(11))+C=\frac(11\cdot\sqrt(x^(5)))(5)+C$.
Substituting the results obtained, we get the answer:
$$5\cdot\int\sqrt(x^4) dx-14\cdot\int\frac(dx)(\sqrt(x^6))= 5\cdot\frac(7\cdot\sqrt(x^( 11)))(11)-14\cdot\frac(11\cdot\sqrt(x^(5)))(5)+C= \frac(35\cdot\sqrt(x^(11)))( 11)-\frac(154\cdot\sqrt(x^(5)))(5)+C. $$
Answer: $\int\left(5\cdot\sqrt(x^4)-\frac(14)(\sqrt(x^6))\right)dx=\frac(35\cdot\sqrt(x^(11 )))(11)-\frac(154\cdot\sqrt(x^(5)))(5)+C$.
And finally, let’s take the integral that falls under formula No. 9 of the table of integrals. Example No. 6, which we will now move on to, could be solved in another way, but this will be discussed in subsequent topics. For now, we will remain within the framework of using the table.
Example No. 6
Find $\int\frac(12)(\sqrt(15-7x^2))dx$.
First, let's do the same operation as before: moving the constant (the number $12$) outside the integral sign:
$$ \int\frac(12)(\sqrt(15-7x^2))dx=12\cdot\int\frac(1)(\sqrt(15-7x^2))dx=12\cdot\int \frac(dx)(\sqrt(15-7x^2)) $$
The resulting integral $\int\frac(dx)(\sqrt(15-7x^2))$ is already close to the tabular one $\int\frac(du)(\sqrt(a^2-u^2))$ (formula No. 9 table of integrals). The difference in our integral is that before $x^2$ under the root there is a coefficient $7$, which the table integral does not allow. Therefore, we need to get rid of this seven by moving it beyond the root sign:
$$ 12\cdot\int\frac(dx)(\sqrt(15-7x^2))=12\cdot\int\frac(dx)(\sqrt(7\cdot\left(\frac(15)() 7)-x^2\right)))= 12\cdot\int\frac(dx)(\sqrt(7)\cdot\sqrt(\frac(15)(7)-x^2))=\frac (12)(\sqrt(7))\cdot\int\frac(dx)(\sqrt(\frac(15)(7)-x^2)) $$
If we compare the table integral $\int\frac(du)(\sqrt(a^2-u^2))$ and $\int\frac(dx)(\sqrt(\frac(15)(7)-x^ 2))$ it becomes clear that they have the same structure. Only in the integral $\int\frac(dx)(\sqrt(\frac(15)(7)-x^2))$ instead of $u$ there is $x$, and instead of $a^2$ there is $\frac (15)(7)$. Well, if $a^2=\frac(15)(7)$, then $a=\sqrt(\frac(15)(7))$. Substituting $u=x$ and $a=\sqrt(\frac(15)(7))$ into the formula $\int\frac(du)(\sqrt(a^2-u^2))=\arcsin\ frac(u)(a)+C$, we get the following result:
$$ \frac(12)(\sqrt(7))\cdot\int\frac(dx)(\sqrt(\frac(15)(7)-x^2))= \frac(12)(\sqrt (7))\cdot\arcsin\frac(x)(\sqrt(\frac(15)(7)))+C $$
If we take into account that $\sqrt(\frac(15)(7))=\frac(\sqrt(15))(\sqrt(7))$, then the result can be rewritten without the “three-story” fractions:
$$ \frac(12)(\sqrt(7))\cdot\arcsin\frac(x)(\sqrt(\frac(15)(7)))+C=\frac(12)(\sqrt(7 ))\cdot\arcsin\frac(x)(\frac(\sqrt(15))(\sqrt(7)))+C= \frac(12)(\sqrt(7))\cdot\arcsin\frac (\sqrt(7)\;x)(\sqrt(15))+C $$
The problem is solved, the answer is received.
Answer: $\int\frac(12)(\sqrt(15-7x^2))dx=\frac(12)(\sqrt(7))\cdot\arcsin\frac(\sqrt(7)\;x) (\sqrt(15))+C$.
Example No. 7
Find $\int\tg^2xdx$.
There are methods for integrating trigonometric functions. However, in this case, you can get by with knowledge of simple trigonometric formulas. Since $\tg x=\frac(\sin x)(\cos x)$, then $\left(\tg x\right)^2=\left(\frac(\sin x)(\cos x) \right)^2=\frac(\sin^2x)(\cos^2x)$. Considering $\sin^2x=1-\cos^2x$, we get:
$$ \frac(\sin^2x)(\cos^2x)=\frac(1-\cos^2x)(\cos^2x)=\frac(1)(\cos^2x)-\frac(\ cos^2x)(\cos^2x)=\frac(1)(\cos^2x)-1 $$
Thus, $\int\tg^2xdx=\int\left(\frac(1)(\cos^2x)-1\right)dx$. Expanding the resulting integral into the sum of integrals and applying tabular formulas, we will have:
$$ \int\left(\frac(1)(\cos^2x)-1\right)dx=\int\frac(dx)(\cos^2x)-\int 1dx=\tg x-x+C . $$
Answer: $\int\tg^2xdx=\tg x-x+C$.
This method comes down to integrating the differential equation of the curved axis of the beam (9.1) with the known law of change in bending moments M(X). Assuming the bending rigidity of the beam to be constant (EJ z= const) and sequentially integrating equation (9.1), we obtain
In expressions (9.5) and below, to simplify the notation, the indices of the moments of inertia and bending moments are omitted.
Expressions (9.5) allow us to obtain analytical laws for changes in deflections and rotation angles in a beam. Integration constants included in (9.5) C 1 and C 2 are subject to determination from the kinematic boundary conditions and the conditions for connecting sections of the beam.
Kinematic boundary conditions reflect the nature of the fastening (support) of the beam and are set relative to deflections and rotation angles. For example, for a simply supported beam (Fig. 9.4), the boundary conditions characterize the absence of deflections on the supports: x = 0, x = /, v = 0. For a cantilever beam (Fig. 9.5), the boundary conditions characterize the equality of the deflection and the angle of rotation in the rigid embedment to zero: x = 0, v= 0; av = 0.
Matching conditions are set at the boundaries of sections with different laws of change in bending moments. In the absence of intermediate hinges and so-called parallelogram mechanisms (sliders), the mating conditions consist in the equality of deflections and rotation angles in the sections to the left and right of the boundary of the sections, that is, they characterize the continuity and smoothness of the curved axis of the beam. For example, for the beam in Fig. 9.4 can be written: X = A, and = and
In the presence of P sections with different laws of change in bending moments, the expression for deflection will contain 2 P integration constants. Using boundary conditions and conditions for connecting sections, we can obtain system 2 P linear algebraic equations with respect to these constants. After determining all integration constants, the laws of change u(x) and ср(х) within each section of the beam will be established. Let's look at examples of determining deflections and rotation angles in beams using the direct integration method.
Example 9.1. Let us determine analytical expressions for u(lc) and cp(x) in a cantilever beam loaded with a uniformly distributed load (Fig. 9.6), and calculate the values of these quantities at the free end.
The bending moment in the beam along its entire length varies according to the law of a square parabola:
Let's substitute this expression into solution (9.5) and integrate it:
Using the boundary conditions, we determine the integration constants:
Let us write down the final expressions for deflections and rotation angles in the beam and determine the values of these quantities at the free end: 
Example 9.2. For a simply supported beam loaded at the end with a concentrated force (Fig. 9.7), we define expressions for y(x) and (p(x) and calculate the values of these quantities in characteristic sections. 
Diagram M shown in Fig. 9.7. Bending moments have different laws of change in the first and second sections of the beam. We integrate the differential equation of the curved axis within each section.
First section (0 2a):
Second section (2 A
To determine the four integration constants C, C 2, D x And D 2 we set boundary conditions and conditions for connecting sections:
From the condition for conjugating the sections, we obtain the equality of the integration constants in the first and second sections: C ( = D v C 2 = D T Using boundary conditions, we find the values of the constants:
Let us write down the final expressions for u(x) and cp(x) within each section: 
In these expressions, a vertical bar with a number at the bottom corresponds to the boundary of each area. Within the first section v and cp are determined by the functions up to the vertical line with the number 1, and within the second section - up to the vertical line with the number 2, that is, by all functions.
Let's calculate v and (p in characteristic sections of the beam:
Within the first section, the sign of the rotation angle changes to the opposite. Let’s set the position of the section where the rotation angle becomes zero:
In section x =x Q the deflection of the beam has an extremum. We calculate its value:
For comparison, we determine the amount of deflection of the beam in the middle of the span:
It can be noted that the extreme deflection differs very slightly (by 2.6%) from the deflection in the middle of the span.
Let's perform a numerical calculation at P= 20 kN and A= 1.6 m. Let us select the section of the beam in the form of a rolled steel I-beam, taking the load reliability factor y^= 1.2, operating conditions coefficient y c = 1, design resistance of the material R= 210 MPa = = 21 kN/cm 2 and modulus of elasticity of steel E- 2.1 10 4 kN/cm 2 .
We accept 120, W z = 184 cm 3, J= 1840 cm 4.
Let's calculate the largest values of the angle of rotation and deflection in the beam. According to SNiP, we carry out calculations based on the effect of standard loads.
From the considered example it is clear that if there are several sections in the beam with different laws of change in bending moments, the direct integration method becomes cumbersome and inconvenient.
Direct integration
Calculation of indefinite integrals using a table of integrals and their basic properties is called direct integration.
Example 1. Let's find the integral
.
Applying the second and fifth properties of the indefinite integral, we obtain
.(*)
Next, using the formulasII, Sh,IV, VIIItables and the third property of integrals, we find each of the terms of the integrals separately:
=
,
,
Let us substitute these results into (*) and, denoting the sum of all constants(3 WITH 1 +7WITH 2 +4WITH 3 +2WITH 4 +WITH 5) letter WITH, we finally get:
Let's check the result by differentiation. Let's find the derivative of the resulting expression:
We have obtained the integrand, this proves that the integration was carried out correctly.
Example 2 . We'll find
.
The table of integrals shows the corollaryIIIA from the formula III:
To use this corollary, we find the differential of a function in the exponent:
To create this differential, it is enough to multiply the denominator of the fraction under the integral by the number 2 (obviously, in order for the fraction not to change, it is necessary to multiply by 2 and numerator). After placing the constant factor outside the integral sign, it becomes ready to apply the tabular formulaIIIA:
.
Examination:
therefore, the integration is done correctly.
Example 3 . We'll find
Since the differential of a quadratic function can be constructed from the expression in the numerator, the following function should be distinguished in the denominator:
.
To create its differential 
it is enough to multiply the numerator by 4 (we also multiply the denominator by 4
and take this factor of the denominator out of the integral). As a result, we will be able to use the tabular formulaX:
Examination:
,
those. the integration was performed correctly.
Example 4 . We'll find
Note that now the quadratic function whose differential 
can be created in the numerator, is a radical expression. Therefore, it would be reasonable to write the integrand as a power function in order to use the formulaItables of integrals:
Examination:
Conclusion: the integral was found correctly.
Example 5. We'll find
Let us note that the integrand contains
function ; and its differential. But the fraction is also the differential of the entire radical expression (up to sign): 
Therefore, it is reasonable to represent the fraction in the form degrees:
Then after multiplying the numerator and denominator by (-1) we get a power integral (tabular formulaI):
By differentiating the result, we make sure that the integration is performed correctly.
Example 6. We'll find
It is easy to see that in this integral from the expression the differential of the radical function cannot be obtained using numerical coefficients. Really,
,
Where k -constant. But, from experience example 3 , it is possible to construct an integral that is identical in form to the formulaXfrom the table of integrals:
Example 7. We'll find
Let us pay attention to the fact that the differential of a cubic function can easily be created in the numeratord(x 3 ) = 3 x 2 dx. After which we get the opportunity to use the tabular formulaVI:
Example 8. We'll find
It is known that the derivative of the function arcsin x is a fraction
Then
.
This leads us to the conclusion that the required integral has the form of a power integral: , in whichand = arcsin x, which means
Example 9 . To find
let's use the same table formula I and the fact that
We get
Example 10 . We'll find
Since the expression is the differential of the function, then, using the formula I tables of integrals, we get
Example 11. To find the integral
let's use sequentially: the trigonometric formula
,
by the fact that
and formula IItables of integrals:
Example 12 . We'll find
.
Since the expression
is the differential of the function , then using the same formulaII, we get
Example 13 . Let's find the integral
Note that the degree of the variable in the numerator is one less than in the denominator. This allows us to create a differential in the numeratordenominator. We'll find
.
After taking the constant factor out of the integral sign, we multiply the numerator and denominator of the integrand by (-7), we get:
(The same formula was used hereIIfrom the table of integrals).
Example 14. Let's find the integral
.
Let's imagine the numerator in a different form: 1 + 2 X 2 = (1 + X 2 )+ x 2 and perform term-by-term division, after which we use the fifth property of integrals and formulasI And VIII tables:
Example 15. We'll find
Let’s take the constant factor beyond the sign of the integral, subtract and add 5 to the numerator, then divide the numerator term by term by the denominator and use the fifth property of the integral:
To calculate the first integral, we use the third property of integrals, and present the second integral in a form convenient for applying the formulaIX:
Example 16. We'll find
Note that the exponent of the variable in the numerator is one less than in the denominator (which is typical for a derivative), which means that the differential of the denominator can be constructed in the numerator. Let's find the differential of the expression in the denominator:
d(x 2- 5)=(X 2 - 5)" dx = 2 xdx.
To obtain a constant factor of 2 in the numerator of the denominator differential, we need to multiply and divide the integrand by 2 and take out the constant factor -
for the integral sign
here we usedIItable integral.
Let's consider a similar situation in the following example.
Example 17. We'll find
.
Let's calculate the differential of the denominator:
.
Let's create it in the numerator using the fourth property of integrals:
=
A more complex similar situation will be considered in example 19.
Example 18 We'll find
.
Let us select a complete square in the denominator:
We get
.
After isolating the perfect square in the denominator, we obtained an integral close in form to the formulasVIII And IXtables of integrals, but in the denominator of the formulaVIIIthe terms of the complete squares have the same signs, and in the denominator of our integral the signs of the terms are different, although they do not coincide with the signs of the ninth formula. Achieve complete coincidence of the signs of the terms in the denominator with the signs in the formulaIXis possible by adding a coefficient (-1) to the integral. So, to apply the formulaIXtables of integrals, we will carry out the following activities:
1) put (-1) outside the brackets in the denominator and then outside the integral;
2) find the differential of the expression
3) create the found differential in the numerator;
4) imagine the number 2 in a form convenient for applying the formulaIX tables:
Then
Using IXformula of the table of integrals, we get
Example 19. We'll find
.
Using the experience gained in finding integrals in the previous two examples and the results obtained in them, we will have
.
Let us summarize some of the experience gained as a result of the solution examples 17,18,19.
So, if we have an integral of the form
(example 18 ), That, by isolating the complete square in the denominator, you can arrive at one of the tabular formulasVIII or IX.
The integral is of the form
(example 19 ) after creating the derivative of the denominator in the numerator, it splits into two integrals: the first one is of the form
( example 17 ), taken from the formulaP, and the second type
(example 18 ), taken from one of the formulasVIII or IX.
Example 20 . We'll find
.
Integral of the form
can be reduced to the form of tabular formulasX or XI, highlighting a complete square in the radical expression. IN in our case
= 
.
The radical expression has the form
The same is always done when calculating integrals of the form
,
if one of the exponents is a positive odd number and the second is an arbitrary real number (example 23 ).
Example 23 . We'll find
Using the experience of the previous example and the identity
2 sin 2 φ = l - cos 2 φ ,2 cos 2 φ = l + cos 2 φ
Substituting the resulting sum into the integral, we get
The problem of finding an antiderivative function does not always have a solution, while we can differentiate any function. This explains the lack of a universal integration method.
In this article we will look at the basic methods for finding the indefinite integral using examples with detailed solutions. We will also group the types of integrand functions characteristic of each integration method.
Page navigation.
Direct integration.
Undoubtedly, the main method of finding an antiderivative function is direct integration using a table of antiderivatives and the properties of the indefinite integral. All other methods are used only to reduce the original integral to tabular form.
Example.
Find the set of antiderivatives of the function.
Solution.
Let's write the function in the form .
Since the integral of a sum of functions is equal to the sum of integrals, then
The numerical coefficient can be taken out of the integral sign: 
The first of the integrals is reduced to tabular form, therefore, from the table of antiderivatives for the exponential function we have 
.
To find the second integral, we use the table of antiderivatives for the power function 
and the rule 
. That is, .
Hence, 
Where
Integration by substitution method.
The essence of the method is that we introduce a new variable, express the integrand through this variable, and as a result we arrive at a tabular (or simpler) form of the integral.
Very often the substitution method comes to the rescue when integrating trigonometric functions and functions with radicals.
Example.
Find the indefinite integral 
.
Solution.
Let's introduce a new variable. Let's express x through z: 
We substitute the resulting expressions into the original integral: 
From the table of antiderivatives we have 
.
It remains to return to the original variable x: 
Answer:
Very often the substitution method is used when integrating trigonometric functions. For example, using the universal trigonometric substitution allows you to transform the integrand into a fractionally rational form.
The substitution method allows you to explain the integration rule .
We introduce a new variable, then
We substitute the resulting expressions into the original integral: 
If we accept and return to the original variable x, we get 
Submitting the differential sign.
The method of subsuming the differential sign is based on reducing the integrand to the form 
. Next, the substitution method is used: a new variable is introduced and after finding the antiderivative for the new variable, we return to the original variable, that is 
For convenience, place it in front of your eyes in the form of differentials to make it easier to convert the integrand, as well as a table of antiderivatives to see what form to convert the integrand to.
For example, let's find the set of antiderivatives of the cotangent function.
Example.
Find the indefinite integral.
Solution.
The integrand can be transformed using trigonometry formulas: 
Looking at the table of derivatives, we conclude that the expression in the numerator can be subsumed under the differential sign 
, That's why 
That is 
.
Let it be then 
. From the table of antiderivatives we see that 
. Returning to the original variable 
.
Without explanation, the solution is written as follows: 
Integration by parts.
Integration by parts is based on representing the integrand as a product and then applying the formula. This method is a very powerful integration tool. Depending on the integrand, the method of integration by parts sometimes has to be applied several times in a row before obtaining the result. For example, let's find the set of antiderivatives of the arctangent function.
Example.
Calculate the indefinite integral.
Solution.
Let it be then 
It should be noted that when finding the function v(x) do not add an arbitrary constant C.
Now we apply the integration by parts formula: 
We calculate the last integral using the method of subsuming it under the differential sign.
Since then 
. That's why 
Hence, 
Where .
Answer:
The main difficulties in integrating by parts arise from the choice: which part of the integrand to take as the function u(x) and which part as the differential d(v(x)). However, there are a number of standard recommendations, which we recommend that you familiarize yourself with in the section integration by parts.
When integrating power expressions, for example or , use recurrent formulas that allow you to reduce the degree from step to step. These formulas are obtained by successive repeated integration by parts. We recommend that you familiarize yourself with the section integration using recurrence formulas.
In conclusion, I would like to summarize all the material in this article. The basis of the fundamentals is the method of direct integration. The methods of substitution, substitution under the differential sign and the method of integration by parts make it possible to reduce the original integral to a tabular one.