May 22, 2021

Chapters

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.

## 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:

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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: