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In this resource, you will find solved examples of triginometric examples. Before proceeding to examples and their solutions, first, let us see the anti derivatives of the common trigonometric functions.

## Common Trigonometric Integrals

The integrals of trigonometric functions are referred to as trigonometric integrals. The integrals of the common trigonometric functions are compiled below:

1. cos x dx = sin x + C

2. sin x dx = -cos x + C

3. x dx = tan x + C

4. x dx = - cotanx + C

5. (sec x tan x)dx = sec x + C

6. (cosec x cotan x)dx = - cosec x + C

The above list is quite helpful in solving the problems related to trigonometric integrals.

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## Example 1

Calculate the following integral:

First, use the sum/difference property of integration to write the above function like this:

Compute the integrals of both the terms separately. The integral of 4 is 4x and the integral of sin x is cos x:

## Example 2

Calculate the following integral:

### Solution

Using the sum/difference property of integration, we can write the above function as:

Calculate the integral or antiderivative of the two terms separately. The antiderivative of 4x is equal to and the antiderivative of 5 is equal to 5x:

E

Example 3

Compute the antiderivative of the following function:

### Solution

can be written as:

According to one of the properties of integrals, we can shift constants before the integral sign. Hence, we can write the above function like this:

We know that csc^2 x dx = - cotan x + C and cos x dx = sin x + C. Hence, we will substitute these values in the above function like this:

## Example 4

Calculate the integral of the following function:

sin (9x + 7) dx

### Solution

In this problem, we will use substitution to compute the integral. Suppose 9x + 7 = u.

If u = 9x + 7, then . This means that dx  is equal to . Now, we will substitute these values in the original function to get the following function:

Shift the fraction before the integral sign:

We know that sin x dx = -cos x + C. Hence, we will substitute this value in the above function:

Since u = 9x + 7, hence we will substitute this value of u in the above function again to get the final answer:

## Example 5

Calculate the integral of the following function:

cos (6x + 1) dx

### Solution

In this problem, we will use substitution to compute the integral. Suppose

If , then . This means that dx  is equal to . Now, we will substitute these values in the original function to get the following function:

Shift the fraction before the integral sign:

We know that cos x dx = sin x + C. Hence, we will substitute this value in the above function:

Since , hence we will substitute this value of u in the above function again to get the final answer:

## Example 6

Calculate the following integral:

### Solution

Suppose , then . It means that is equal to .

Shift fraction on the left side of the integral to get:

Remember that the antiderivative of sin u is equal to - cos u + C. Hence, we can write the function as:

Put to get the following answer:

## Example 7

Calculate the following integral:

### Solution

Suppose , then . It means that is equal to .

Shift fraction on the left side of the integral to get:

Remember that the antiderivative of cos u is equal to sin u + C. Hence, we can write the function as:

Put to get the following answer:

## Example 8

Calculate the integral of the following function:

### Solution

We can write as the product of and sin x like this:

We know that . This means that is equal to .

Substitute u = cos x:

is equal to :

Substitute :

## Example 9

Calculate the integral of the following function:

tan (3x - 4) dx

### Solution

In this problem, we will use substitution to compute the integral. Suppose

If , then . This means that dx  is equal to . Now, we will substitute these values in the original function to get the following function:

Shift the fraction before the integral sign:

We know that tan x dx = - ln |cos x| + C. Hence, we will substitute this value in the above function:

Since , hence we will substitute this value of u in the above function again to get the final answer:

## Example 10

Evaluate the following function:

### Solution

As the power of sin x is odd, hence we can write the above function like this:

Substitute in the above function:

Suppose u = cos x, then du = -sin x:

Integrate the above function like this:

Substitute u = cos x again in the above function:

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