# Trigonometric integrals Calculator

## Get detailed solutions to your math problems with our Trigonometric integrals step by step calculator. Sharpen your math skills and learn step by step with our math solver. Check out more online calculators here.

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### Difficult Problems

1

Solved example of Integration by trigonometric substitution

$\int\sqrt{x^2+4}dx$
2

Solve the integral $\int\sqrt{x^2+4}dx$ by trigonometric substitution using the substitution

$\begin{matrix}x=2\tan\left(\theta\right) \\ dx=2\sec\left(\theta\right)^2d\theta\end{matrix}$
3

Substituting in the original integral, we get

$\int2\sqrt{4\tan\left(\theta\right)^2+4}\sec\left(\theta\right)^2d\theta$
4

Take the constant out of the integral

$2\int\sqrt{4\tan\left(\theta\right)^2+4}\sec\left(\theta\right)^2d\theta$
5

Factor by the greatest common divisor $4$

$2\int\sqrt{4\left(\tan\left(\theta\right)^2+1\right)}\sec\left(\theta\right)^2d\theta$
6

The power of a product is equal to the product of it's factors raised to the same power

$2\int2\sqrt{\tan\left(\theta\right)^2+1}\sec\left(\theta\right)^2d\theta$
7

Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$

$2\int2\sec\left(\theta\right)^2\sec\left(\theta\right)d\theta$
8

Take the constant out of the integral

$4\int\sec\left(\theta\right)^2\sec\left(\theta\right)d\theta$
9

When multiplying exponents with same base you can add the exponents

$4\int\sec\left(\theta\right)^{3}d\theta$
10

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$4\left(\frac{\sec\left(\theta\right)^{2}\sin\left(\theta\right)}{2}+\frac{1}{2}\int\sec\left(\theta\right)d\theta\right)$
11

Expressing the result of the integral in terms of the original variable

$4\left(\frac{\frac{\left(\frac{\sqrt{x^2+4}}{2}\right)^{2}x}{\sqrt{x^2+4}}}{2}+\frac{1}{2}\int\sec\left(\theta\right)d\theta\right)$
12

Simplifying the fraction

$4\left(\frac{\left(\frac{\sqrt{x^2+4}}{2}\right)^{2}x}{2\sqrt{x^2+4}}+\frac{1}{2}\int\sec\left(\theta\right)d\theta\right)$
13

Take out the constant $2$ from the fraction's denominator

$4\left(\frac{1}{2}\left(\frac{\left(\frac{\sqrt{x^2+4}}{2}\right)^{2}x}{\sqrt{x^2+4}}\right)+\frac{1}{2}\int\sec\left(\theta\right)d\theta\right)$
14

The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$

$4\left(\frac{1}{2}\left(\frac{\left(\frac{\sqrt{x^2+4}}{2}\right)^{2}x}{\sqrt{x^2+4}}\right)+\frac{1}{2}\ln\left|\sec\left(\theta\right)+\tan\left(\theta\right)\right|\right)$
15

Expressing the result of the integral in terms of the original variable

$4\left(\frac{\frac{1}{2}\left(\frac{\sqrt{x^2+4}}{2}\right)^{2}x}{\sqrt{x^2+4}}+\frac{1}{2}\ln\left|\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right|\right)$
16

Add fraction's numerators with common denominators: $\frac{\sqrt{x^2+4}}{2}$ and $\frac{x}{2}$

$4\left(\frac{\frac{1}{2}\left(\frac{\sqrt{x^2+4}}{2}\right)^{2}x}{\sqrt{x^2+4}}+\frac{1}{2}\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|\right)$
17

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

$4\left(\frac{\frac{1}{2}\left(\frac{\sqrt{x^2+4}}{2}\right)^{2}x}{\sqrt{x^2+4}}+\frac{1}{2}\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|\right)+C_0$