Table of basic integrals

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

Direct integration is called the evaluation of indefinite integrals by bringing them to the tabulated integrals using the basic properties.

Problem 3. Find the indefinite integral and the result of integration verify by the differentiation.

Solution. We apply the method of direct integration.

.

Verification. .

We got the integrand, hence the integral is found correctly.

Problem 4. Find indefinite integrals:

4.1

4.2

Solution. 4.1 . We apply the method of integration by parts. The formula of integration by parts is as follows:

(1)

Remark 1. From the considered integral function u and expression dv are chosen so that is simpler than the original integral.

Remark 2. We can find integrals of functions of the forms: by integrating by parts: take multiplier as u and use formula (1) n times.

Suppose, , then .

According to the formula (1) we find:

.

The last integral again integrate by parts. Suppose , then

.

Finally, we have

.

Solution. 4.2 . We apply also the method of integration by parts. Suppose

, then .

According to the formula (1) we find:

.

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