# Integrals of Trigonometric Functions with Examples

Integrals of trigonometric functions are other trigonometric functions. For example, the integral of the cosine function is equal to the sine function and the integral of the sine function is equal to negative cosine.

Here, we will look at the most important formulas for the integrals of trigonometric functions. Then, we will apply these formulas to solve some practice problems.

##### CALCULUS

Relevant for

Learning about integrals of trigonometric functions.

See formulas

##### CALCULUS

Relevant for

Learning about integrals of trigonometric functions.

See formulas

## Formulas for integrals of trigonometric functions

### Integral of the sine function

The integral of the standard sine function is:

$$\int \sin(x) dx=-\cos(x)+c$$

The integral of the sine function of an angle of the form $latex nx$ is:

$$\int \sin(nx) dx=-\frac{1}{n}\cos(nx)+c$$

We can integrate compositions of the sine function, such as $latex \sin(2x)$ or $latex \sin^2(x)$ using the chain rule for integrals.

### Integral of the cosine function

The integral of the standard cosine function is:

$$\int \cos(x) dx=\sin(x)+c$$

The integral of the cosine function of an angle of the form $latex nx$ is:

$$\int \cos(nx) dx=\frac{1}{n}\sin(nx)+c$$

### Integral of the tangent function

The integral of the standard tangent function is:

$$\int \tan(x) dx=\ln| \sec(x)|+c$$

The integral of the tangent function multiplied by the secant is:

$$\int \sec(x) \tan(x) dx=\sec(x)+c$$

### Integral of the cosecant function

The integral of the standard cosecant function is:

$$\int \cosec(x) dx=\ln\left| \tan\left(\frac{x}{2} \right)\right| +c$$

The integral of the cosecant squared function:

$$\int \cosec^2(x) dx=-\cot(x) +c$$

### Integral of the secant function

The integral of the standard secant function is:

$$\int \sec(x) dx=\ln| \tan(x)+\sec(x)| +c$$

The integral of the secant squared function is:

$$\int \sec^2(x) dx=\tan(x) +c$$

### Integral of the cotangent function

The integral of the standard cotangent function is:

$$\int \cot(x)dx=\ln|\sin(x)|+c$$

The integral of the cotangent function multiplied by the cosecant is:

$$\int \cosec(x) \cot(x)dx=-\cosec(x)+c$$

## Integrals of trigonometric functions – Examples with answers

### EXAMPLE 1

Solve the following integral:

$$\int \sin(4x) dx$$

We can use the chain rule for integration to solve this integral.

Then, we know that the integral of $latex \sin(x)$ is equal to $latex -\cos(x)$. Also, we note that the derivative of $latex 4x$ is 4, so we have:

$$\int \sin(4x) dx=-\frac{1}{4}\cos(4x)+c$$

### EXAMPLE 2

Solve the following integral:

$$\int \sin(x) \cos(x)dx$$

To solve this integral, we can use the following trigonometric identity:

$$\sin(2x) \equiv 2 \sin(x) \cos(x)$$

$$\sin(x)\cos(x) \equiv \frac{1}{2}\sin(2x)$$

Then, we have:

$$\int \sin(x) \cos(x)dx =\int \frac{1}{2}\sin(2x) dx$$

$$=\frac{1}{2}\left(-\frac{1}{2} \cos(2x)\right)+c$$

$$=-\frac{1}{4}\cos(2x)+c$$

### EXAMPLE 3

Find the following integral:

$$\int x^2 \cos(x^3-2)dx$$

We can solve this integral by observing that the derivative of $latex x^3-2$ is $latex 3x^2$ and that we have a term $latex x^2$ outside the main function.

Then, we use the chain rule for integrals and we have:

$$\int x^2 \cos(x^3-2)dx=\frac{1}{3}\sin(x^3-2)+c$$

### EXAMPLE 4

Find the following integral:

$$\int \cos(x) \sin^2(x)dx$$

This integral is found by observing that the derivative of $latex \sin(x)$ is $latex \cos(x)$ and that the function $latex \cos(x)$ is outside the main function.

Then, we have:

$$\int \cos(x) \sin^2(x)dx =\frac{(\sin(x))^3}{3}+c$$

$$=\frac{\sin^3(x)}{3}+c$$

### EXAMPLE 5

Find the following integral:

$$\int 2\sec(3x) \tan(3x)dx$$

To solve this integral, we are going to use the formula $latex \int \sec(x) \tan(x) dx=\sec(x)+c$.

In addition, we observe that the derivative of $latex 3x$ is 3:

$$\int 2\sec(3x) \tan(3x)dx=\frac{2}{3}\sec(3x)+c$$

### EXAMPLE 6

What is the result of the following integral?

$$\int x\sec^2(1-x^2) dx$$

We are going to use the integral $$int \sec^2(x) dx=\tan(x) +c$$ to solve this problem.

Then, we observe that the derivative of $latex (1-x^2)$ is $latex -2x$, and we that have a term $latex x$ outside the main function. Then, we have:

$$\int x\sec^2(1-x^2) dx=\frac{\tan(1-x^2)}{2}+c$$

## Integrals of trigonometric functions – Practice problems

Integrals of trigonometric functions quiz
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#### Find the integral $latex \int 12\cosec(4x)\cot(4x) dx$

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$latex ~~=$