Welcome to our page of solved integral exercises! If you are a student, you should know that integral calculus is one of the most important areas of mathematics with countless applications in other areas of knowledge.

In this article, you will find step-by-step explanations, solved examples, and useful tips for solving integrals effectively using different integration techniques. Whether you are looking to improve your mathematical skills or simply need help with a specific problem, you've come to the right place!

We invite you to solve the following integrals yourself, and then check your answers with the collapsible solutions that Superprof has for you. Let's go!

Solve the following integrals

1

Solution

To solve the integral, we move the denominator up and simplify the powers, then apply the immediate power integral.

 

2

Solution

To solve the integral, we make the variable substitution and then apply the immediate power integral.

3

Solution

To solve the integral, we make the variable substitution and then apply the immediate power integral.

4

Solution

To solve the integral, we make the variable substitution and then apply the immediate power integral.

 

5

Solution

To solve the integral, we make the variable substitution and then apply the immediate power integral.

6

Solution

To solve the integral, we make the variable substitution and then apply the immediate power integral.


7

Solution

To solve the integral, we make the variable substitution and then apply the immediate power integral.

8

Solution

To solve the integral, we make the variable substitution and then apply the immediate power integral.

9

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

10

Solution

To solve the integral, we use the definition of tangent in terms of sine and cosine, then apply the immediate integral .

11

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

12

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

13

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

14

Solution

To solve the integral, we separate the integral and then apply the immediate integral .

15

Solution

To solve the integral, we add a zero to be able to separate it into two integrals and then apply the immediate integral .


16

Solution

To solve the integral, we start by doing synthetic division to be able to separate it into two integrals and then apply the immediate integral , with a variable substitution .

 

17

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

18

Solution

To solve the integral, we apply the immediate integral .

19

Solution

To solve the integral, we apply the immediate integral .

20

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

21

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

22

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

23

Solution

To solve the integral, we make the variable substitution and then we apply the immediate integral .

24

Solution

We start by separating the integral and applying the respective immediate integrals

25

Solution

We start by making a variable substitution, and applying the immediate integral .

26

Solution

We start by making a variable substitution, and applying the immediate integral .

 

27

Solution

We start by making a variable substitution, and applying the immediate integral .

28

Solution

We start by making a variable substitution, and applying the immediate integral .

29

Solution

We start by using the identity , separating the integral and applying the immediate integral , with a variable substitution .

30

Solution

We start by separating the sine and using the trigonometric identity , applying the immediate integral and a variable substitution

 

31

Solution

We start by separating the integral and applying the immediate integral

32

Solution

We start by making a variable substitution and applying the immediate integral

33

Solution

We start by making a variable substitution and applying the immediate integral

34

Solution

We start by making a variable substitution and applying the immediate integral

35

Solution

We start by using the trigonometric identity , separating the integral and applying the immediate integral , with a variable substitution

 

36

Solution

We start by using the trigonometric identity , separating the integral, applying the immediate integral , and a variable substitution

 

37

Solution

We start by using the trigonometric identity and applying the immediate integral }

38

Solution

We start by using the trigonometric identity and applying the immediate integral }

39

Solution

We start by making a variable substitution and applying the immediate integral }

40

Solution

We start by separating the secant and using the trigonometric identity , applying the immediate integral and a variable substitution

 

41

Solution

We start by using the trigonometric identity and applying the immediate integral

42

Solution

We start by adding a zero, using the trigonometric identity , separating the integral and applying the immediate integral

43

Solution

To solve the integral we are going to find the values of A and B that satisfy the following identity

 

 

We apply the previous identity, a variable substitution and apply the integral

 

44

Solution

To solve the integral we use the trigonometric identity , separate the integral and simplify,

 


Now we use the definitions and and finally the immediate integrals and

45

Solution

To solve the following integral we multiply the numerator and denominator by , and then separate the integral


We apply the integral and the variable substitution

 

46

Solution

To solve the following integral we multiply the numerator and denominator by , expand the squared binomial, use the identity and then separate the integral


Now we use the definitions , the immediate integral , a variable substitution and add a zero, to be able to use the identity and finally the integral

 

 

 

47

Solution

To solve the following integral we multiply the numerator and denominator by , expand the squared binomial, use the identity and then separate the integral

 


Now we use the definitions , the immediate integral , a variable substitution and add a zero, to be able to use the identity

 

 

48

Solution

To solve the following integral we look to complete squares to have an integral of the form , where .

 

 

49

Solution

We start by separating the integral, on one hand we have a variable substitution , on the other hand we look to have an integral of the form , where .

 

 

50

Solution

We look to have an integral of the form , where .

 

51

Solution

We look to have an integral of the form , so in the denominator we can make the following substitution:


donde .

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.