# Integral of x/((2^2-1x^2)^0.5)

## \int\frac{x}{\sqrt{2^2-x^2}}dx

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ln
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asin
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asinh
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$-\sqrt{4-x^2}+C_0$

## Step by step solution

Problem

$\int\frac{x}{\sqrt{2^2-x^2}}dx$
1

Calculate the power

$\int\frac{x}{\sqrt{4-x^2}}dx$
2

Solve the integral $\int\frac{x}{\sqrt{4-x^2}}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=4-x^2 \\ du=-2xdx\end{matrix}$
3

Isolate $dx$ in the previous equation

$\frac{du}{-2x}=dx$
4

Substituting $u$ and $dx$ in the integral

$\int\frac{1}{-2\sqrt{u}}du$
5

Taking the constant out of the integral

$-\frac{1}{2}\int\frac{1}{\sqrt{u}}du$
6

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$-\frac{1}{2}\int u^{-\frac{1}{2}}du$
7

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$-\frac{1}{2}\cdot 2\sqrt{u}$
8

Substitute $u$ back for it's value, $4-x^2$

$-\sqrt{4-x^2}$
9

$-\sqrt{4-x^2}+C_0$

$-\sqrt{4-x^2}+C_0$

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### Main topic:

Integration by substitution

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