# Integration by trigonometric substitution Calculator

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### Difficult Problems

1

Example

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

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

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

Take the constant out of the integral

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

Factor by the greatest common divisor $4$

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

Any expression multiplied by $1$ is equal to itself

$2\int\sqrt{4\left(1+\tan\left(\theta\right)^2\right)}\sec\left(\theta\right)^2d\theta$
7

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

$2\int2\sec\left(\theta\right)^2\sqrt{1+\tan\left(\theta\right)^2}d\theta$
8

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

$2\int2\sec\left(\theta\right)^2\sec\left(\theta\right)d\theta$
9

When multiplying exponents with same base you can add the exponents

$2\int2\sec\left(\theta\right)^{3}d\theta$
10

Take the constant out of the integral

$4\int\sec\left(\theta\right)^{3}d\theta$
11

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

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

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

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

Simplifying the fraction

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

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

$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{1}{2}\cdot\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
15

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

$4\left(\frac{1}{2}\ln\left(\tan\left(\theta\right)+\sec\left(\theta\right)\right)+\frac{1}{2}\cdot\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
16

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

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

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

$4\left(\frac{1}{2}\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
18

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

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}$
19

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{2}x\left(\frac{4+x^2}{4}\right)}{\sqrt{4+x^2}}$
20

Simplify the fraction

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{8}x\left(4+x^2\right)}{\sqrt{4+x^2}}$
21

Simplifying the fraction by $4+x^2$

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{8}x}{\frac{1}{\sqrt{4+x^2}}}$
22

Simplifying the fraction

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\cdot \frac{1}{8}\sqrt{4+x^2}x$
23

Multiply $\frac{1}{8}$ times $4$

$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+\frac{1}{2}\sqrt{4+x^2}x$
24

The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator

$2\left(\ln\left(2\right)\left(-1\right)+\ln\left(x+\sqrt{4+x^2}\right)\right)+\frac{1}{2}\sqrt{4+x^2}x$
25

Calculating the natural logarithm of $2$

$2\left(\frac{4}{\sqrt{3}}\left(-1\right)+\ln\left(x+\sqrt{4+x^2}\right)\right)+\frac{1}{2}\sqrt{4+x^2}x$
26

Multiply $-1$ times $\frac{4}{\sqrt{3}}$

$2\left(\ln\left(x+\sqrt{4+x^2}\right)-\frac{4}{\sqrt{3}}\right)+\frac{1}{2}\sqrt{4+x^2}x$
27

Multiply $\left(\ln\left(x+\sqrt{4+x^2}\right)+-\frac{4}{\sqrt{3}}\right)$ by $2$

$\frac{1}{2}\sqrt{4+x^2}x-\frac{17}{\sqrt{3}}+2\ln\left(x+\sqrt{4+x^2}\right)$
28

$\frac{1}{2}\sqrt{4+x^2}x-\frac{17}{\sqrt{3}}+2\ln\left(x+\sqrt{4+x^2}\right)+C_0$