# Integral of x/((x^2+9)^0.5)

## \int\frac{x}{\sqrt{x^2+9}}dx

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ln
log
lim
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sin
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csc

asin
acos
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sinh
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tanh
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csch

asinh
acosh
atanh
acoth
asech
acsch

$\sqrt{9+x^2}+C_0$

## Step by step solution

Problem

$\int\frac{x}{\sqrt{x^2+9}}dx$
1

Solve the integral $\int\frac{x}{\sqrt{9+x^2}}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=9+x^2 \\ du=2xdx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{2x}=dx$
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Substituting $u$ and $dx$ in the integral

$\int\frac{1}{2\sqrt{u}}du$
4

Taking the constant out of the integral

$\frac{1}{2}\int\frac{1}{\sqrt{u}}du$
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Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\frac{1}{2}\int u^{-\frac{1}{2}}du$
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Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{1}{2}\cdot 2\sqrt{u}$
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Substitute $u$ back for it's value, $9+x^2$

$1\sqrt{9+x^2}$
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Any expression multiplied by $1$ is equal to itself

$\sqrt{9+x^2}$
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$\sqrt{9+x^2}+C_0$

$\sqrt{9+x^2}+C_0$

### Main topic:

Integration by substitution

0.5 seconds

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