# Integral of 1/((1+x^2)^3^0.5)

## \int\frac{1}{\sqrt{\left(1+x^2\right)^3}}dx

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$\frac{x}{\sqrt{x^2+1}}+C_0$

## Step by step solution

Problem

$\int\frac{1}{\sqrt{\left(1+x^2\right)^3}}dx$
1

Applying the power of a power property

$\int\frac{1}{\sqrt{\left(x^2+1\right)^{3}}}dx$
2

Solve the integral $\int\frac{1}{\sqrt{\left(x^2+1\right)^{3}}}$ by trigonometric substitution using the substitution

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

Substituting in the original integral, we get

$\int\frac{\sec\left(\theta\right)^2}{\sqrt{\left(\tan\left(\theta\right)^2+1\right)^{3}}}d\theta$
4

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

$\int\frac{\sec\left(\theta\right)^2}{\sec\left(\theta\right)^{3}}d\theta$
5

Simplifying the fraction by $\sec\left(\theta\right)$

$\int\sec\left(\theta\right)^{\left(2-3\right)}d\theta$
6

Subtract the values $2$ and $-3$

$\int\sec\left(\theta\right)^{-1}d\theta$
7

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\int\frac{1}{\sec\left(\theta\right)}d\theta$
8

Applying the trigonometric identity: $\displaystyle\frac{1}{\sec(\theta)}=\cos(\theta)$

$\int\cos\left(\theta\right)d\theta$
9

Apply the integral of the cosine function

$\sin\left(\theta\right)$
10

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

$\frac{x}{\sqrt{x^2+1}}$
11

$\frac{x}{\sqrt{x^2+1}}+C_0$

$\frac{x}{\sqrt{x^2+1}}+C_0$

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### Main topic:

Integration by trigonometric substitution

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