Integration by parts. Examples of solutions

Hello again. Today in the lesson we will learn how to integrate by parts. The method of integration by parts is one of the cornerstones of integral calculus. During tests or exams, students are almost always asked to solve the following types of integrals: the simplest integral (see article) or an integral by replacing a variable (see article) or the integral is just on integration by parts method.

As always, you should have on hand: Table of integrals And Derivatives table. If you still don’t have them, then please visit the storage room of my website: Mathematical formulas and tables. I won’t tire of repeating – it’s better to print everything out. I will try to present all the material consistently, simply and clearly; there are no particular difficulties in integrating the parts.

What problem does the method of integration by parts solve? The method of integration by parts solves a very important problem; it allows you to integrate some functions that are not in the table, work functions, and in some cases – even quotients. As we remember, there is no convenient formula: . But there is this one: – formula for integration by parts in person. I know, I know, you’re the only one - we’ll work with her throughout the lesson (it’s easier now).

And immediately the list is sent to the studio. The integrals of the following types are taken by parts:

1) , , – logarithm, logarithm multiplied by some polynomial.

2) ,is an exponential function multiplied by some polynomial. This also includes integrals like - an exponential function multiplied by a polynomial, but in practice this is 97 percent, under the integral there is a nice letter “e”. ... the article turns out to be somewhat lyrical, oh yes ... spring has come.

3) , , are trigonometric functions multiplied by some polynomial.

4) , – inverse trigonometric functions (“arches”), “arches” multiplied by some polynomial.

Some fractions are also taken in parts; we will also consider the corresponding examples in detail.

Integrals of logarithms

Example 1

Classic. From time to time this integral can be found in tables, but it is not advisable to use a ready-made answer, since the teacher has spring vitamin deficiency and will swear heavily. Because the integral under consideration is by no means tabular - it is taken in parts. We decide:

We interrupt the solution for intermediate explanations.

We use the integration by parts formula:

The formula is applied from left to right

We look at the left side: . Obviously, in our example (and in all the others that we will consider), something needs to be designated as , and something as .

In integrals of the type under consideration, the logarithm is always denoted.

Technically, the design of the solution is implemented as follows; we write in the column:

That is, we denoted the logarithm by, and by - the remaining part integrand expression.

Next stage: find the differential:

A differential is almost the same as a derivative; we have already discussed how to find it in previous lessons.

Now we find the function. In order to find the function you need to integrate right side lower equality:

Now we open our solution and construct the right side of the formula: .
By the way, here is a sample of the final solution with some notes:

The only point in the work is that I immediately swapped and , since it is customary to write the factor before the logarithm.

As you can see, applying the integration by parts formula essentially reduced our solution to two simple integrals.

Please note that in some cases right after application of the formula, a simplification is necessarily carried out under the remaining integral - in the example under consideration, we reduced the integrand to “x”.

Let's check. To do this, you need to take the derivative of the answer:

The original integrand function has been obtained, which means that the integral has been solved correctly.

During the test, we used the product differentiation rule: . And this is no coincidence.

Formula for integration by parts and formula – these are two mutually inverse rules.

Example 2

Find the indefinite integral.

The integrand is the product of a logarithm and a polynomial.
Let's decide.

I will once again describe in detail the procedure for applying the rule; in the future, examples will be presented more briefly, and if you have difficulties in solving it on your own, you need to go back to the first two examples of the lesson.

As already mentioned, it is necessary to denote the logarithm (the fact that it is a power does not matter). We denote by the remaining part integrand expression.

We write in the column:

First we find the differential:

Here we use the rule for differentiating a complex function . It is no coincidence that at the very first lesson of the topic Indefinite integral. Examples of solutions I focused on the fact that in order to master integrals, it is necessary to “get your hands on” derivatives. You will have to deal with derivatives more than once.

Now we find the function, for this we integrate right side lower equality:

For integration we used the simplest tabular formula

Now everything is ready to apply the formula . Open with an asterisk and “construct” the solution in accordance with the right side:

Under the integral we again have a polynomial for the logarithm! Therefore, the solution is again interrupted and the rule of integration by parts is applied a second time. Do not forget that in similar situations the logarithm is always denoted.

It would be good if by now you knew how to find the simplest integrals and derivatives orally.

(1) Don't get confused about the signs! Very often the minus is lost here, also note that the minus refers to to all bracket , and these brackets need to be expanded correctly.

(2) Open the brackets. We simplify the last integral.

(3) We take the last integral.

(4) “Combing” the answer.

The need to apply the rule of integration by parts twice (or even three times) does not arise very rarely.

And now a couple of examples for your own solution:

Example 3

Find the indefinite integral.

This example is solved by changing the variable (or substituting it under the differential sign)! Why not - you can try taking it in parts, it will turn out to be a funny thing.

Example 4

Find the indefinite integral.

But this integral is integrated by parts (the promised fraction).

These are examples for you to solve on your own, solutions and answers at the end of the lesson.

It seems that in examples 3 and 4 the integrands are similar, but the solution methods are different! This is the main difficulty in mastering integrals - if you choose the wrong method for solving an integral, then you can tinker with it for hours, like with a real puzzle. Therefore, the more you solve various integrals, the better, the easier the test and exam will be. In addition, in the second year there will be differential equations, and without experience in solving integrals and derivatives there is nothing to do there.

In terms of logarithms, this is probably more than enough. As an aside, I can also remember that engineering students use logarithms to call female breasts =). By the way, it is useful to know by heart the graphs of the main elementary functions: sine, cosine, arctangent, exponent, polynomials of the third, fourth degree, etc. No, of course, a condom on the globe
I won’t stretch it, but now you will remember a lot from the section Charts and functions =).

Integrals of an exponential multiplied by a polynomial

General rule:

Example 5

Find the indefinite integral.

Using a familiar algorithm, we integrate by parts:

If you have difficulties with the integral, then you should return to the article Variable change method in indefinite integral.

The only other thing you can do is tweak the answer:

But if your calculation technique is not very good, then the most profitable option is to leave it as an answer or even

That is, the example is considered solved when the last integral is taken. It won’t be a mistake; it’s another matter that the teacher may ask you to simplify the answer.

Example 6

Find the indefinite integral.

This is an example for you to solve on your own. This integral is integrated twice by parts. Particular attention should be paid to the signs - it’s easy to get confused in them, we also remember that this is a complex function.

There is nothing more to say about the exhibitor. I can only add that the exponential and the natural logarithm are mutually inverse functions, this is me on the topic of entertaining graphs of higher mathematics =) Stop, stop, don’t worry, the lecturer is sober.

Integrals of trigonometric functions multiplied by a polynomial

General rule: for always denotes a polynomial

Example 7

Find the indefinite integral.

Let's integrate by parts:

Hmmm, ...and there is nothing to comment on.

Example 8

Find the indefinite integral

This is an example for you to solve yourself

Example 9

Find the indefinite integral

Another example with a fraction. As in the two previous examples, for denotes a polynomial.

Let's integrate by parts:

If you have any difficulties or misunderstandings with finding the integral, I recommend attending the lesson Integrals of trigonometric functions.

Example 10

Find the indefinite integral

This is an example for you to solve on your own.

Hint: Before using the integration by parts method, you should apply some trigonometric formula that turns the product of two trigonometric functions into one function. The formula can also be used when applying the method of integration by parts, whichever is more convenient for you.

That's probably all in this paragraph. For some reason I remembered a line from the physics and mathematics hymn “And the sine graph runs wave after wave along the abscissa axis”….

Integrals of inverse trigonometric functions.
Integrals of inverse trigonometric functions multiplied by a polynomial

General rule: always denotes the inverse trigonometric function.

Let me remind you that the inverse trigonometric functions include arcsine, arccosine, arctangent and arccotangent. For the sake of brevity of the record I will call them "arches"

Examples of solutions of integrals by parts, the integrand of which contains the logarithm, arcsine, arctangent, as well as the logarithm to the integer power and the logarithm of the polynomial, are considered in detail.

See also: Method of integration by parts
Table of indefinite integrals
Methods for calculating indefinite integrals
Basic elementary functions and their properties

Formula for integration by parts

Below, when solving examples, the integration by parts formula is used:
;
.

Examples of integrals containing logarithms and inverse trigonometric functions

Here are examples of integrals that are integrated by parts:
, , , , , , .

When integrating, that part of the integrand that contains the logarithm or inverse trigonometric functions is denoted by u, the rest by dv.

Below are examples with detailed solutions of these integrals.

Simple example with logarithm

Let's calculate the integral containing the product of a polynomial and a logarithm:

Here the integrand contains a logarithm. Making substitutions
u = ln x, dv = x 2 dx . Then
,
.

Let's integrate by parts.
.

.
Then
.
At the end of the calculations, add the constant C.

Example of a logarithm to the power of 2

Let's consider an example in which the integrand includes a logarithm to an integer power. Such integrals can also be integrated by parts.

Making substitutions
u = (ln x) 2, dv = x dx . Then
,
.

We also calculate the remaining integral by parts:
.
Let's substitute
.

An example in which the logarithm argument is a polynomial

Integrals can be calculated by parts, the integrand of which includes a logarithm whose argument is a polynomial, rational or irrational function. As an example, let's calculate an integral with a logarithm whose argument is a polynomial.
.

Making substitutions
u = ln( x 2 - 1), dv = x dx .
Then
,
.

We calculate the remaining integral:
.
We do not write the modulus sign here ln | x 2 - 1|, since the integrand is defined at x 2 - 1 > 0 . Let's substitute
.

Arcsine example

Let's consider an example of an integral whose integrand includes the arcsine.
.

Making substitutions
u = arcsin x,
.
Then
,
.

Next, we note that the integrand is defined for |x|< 1 . Let us expand the sign of the modulus under the logarithm, taking into account that 1 - x > 0 And 1 + x > 0.

Arc tangent example

Let's solve the example with arctangent:
.

Let's integrate by parts.
.
Let's select the whole part of the fraction:
x 8 = x 8 + x 6 - x 6 - x 4 + x 4 + x 2 - x 2 - 1 + 1 = (x 2 + 1)(x 6 - x 4 + x 2 - 1) + 1;
.
Let's integrate:
.
Finally we have.

Integrals of logarithms

Integration by parts. Examples of solutions

Solution.

Eg.

Calculate the integral:

Using the properties of the integral (linearity), ᴛ.ᴇ. , we reduce it to a tabular integral, we get that

Hello again. Today in the lesson we will learn how to integrate by parts. The method of integration by parts is one of the cornerstones of integral calculation. During tests or exams, students are almost always asked to solve the following types of integrals: the simplest integral (see articleIndefinite integral. Examples of solutions ) or an integral by replacing a variable (see articleVariable change method in an indefinite integral ) or the integral is just on integration by parts method.

As always, you should have at hand: Table of integrals And Derivatives table. If you still don’t have them, then please visit the storeroom of my website: Mathematical formulas and tables. I won’t tire of repeating – it’s better to print everything out. I will try to present all the material consistently, simply and clearly; there are no particular difficulties in integrating the parts.

What problem does the method of integration by parts solve? The method of integration by parts solves a very important problem; it allows you to integrate some functions that are not in the table, work functions, and in some cases – even quotients. As we remember, there is no convenient formula: . But there is this: - the formula for integration by parts in person. I know, I know, you’re the only one - we’ll work with her throughout the lesson (it’s easier now).

And immediately the list is sent to the studio. The integrals of the following types are taken by parts:

1) , – logarithm, logarithm multiplied by some polynomial.

2) , is an exponential function multiplied by some polynomial. This also includes integrals like - an exponential function multiplied by a polynomial, but in practice this is 97 percent, under the integral there is a nice letter ʼʼеʼʼ. ... the article turns out to be somewhat lyrical, oh yes ... spring has come.

3) , are trigonometric functions multiplied by some polynomial.

4) , – inverse trigonometric functions (‘arches’), ‘arches’, multiplied by some polynomial.

Some fractions are also taken in parts; we will also consider the corresponding examples in detail.

Example 1

Find the indefinite integral.

Classic. From time to time this integral can be found in tables, but it is not advisable to use a ready-made answer, since the teacher has spring vitamin deficiency and will swear heavily. Because the integral under consideration is by no means tabular - it is taken in parts. We decide:

We interrupt the solution for intermediate explanations.

We use the integration by parts formula:

Integrals of logarithms - concept and types. Classification and features of the category "Integrals of logarithms" 2017, 2018.