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.
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:
$$ \int \sin(4x) dx$$
Solution
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:
$$ \int \sin(x) \cos(x)dx$$
Solution
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:
$$ \int x^2 \cos(x^3-2)dx$$
Solution
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:
$$ \int \cos(x) \sin^2(x)dx$$
Solution
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:
$$ \int 2\sec(3x) \tan(3x)dx$$
Solution
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?
$$ \int x\sec^2(1-x^2) dx$$
Solution
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
Find the integral $latex \int 12\cosec(4x)\cot(4x) dx$
Write the answer in the input box.
See also
Interested in learning more about integrals? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
