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

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

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Answer

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

Step by step solution

Problem

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

Taking the constant out of the integral

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

Solve the integral $\int\frac{x}{\sqrt{1+x^2}}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=1+x^2 \\ du=2xdx\end{matrix}$
3

Isolate $dx$ in the previous equation

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

Substituting $u$ and $dx$ in the integral

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

Taking the constant out of the integral

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

Multiply $\frac{1}{2}$ times $2$

$1\int\frac{1}{\sqrt{u}}du$
7

Any expression multiplied by $1$ is equal to itself

$\int\frac{1}{\sqrt{u}}du$
8

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

$\int u^{-\frac{1}{2}}du$
9

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

$2\sqrt{u}$
10

Substitute $u$ back for it's value, $1+x^2$

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

Add the constant of integration

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

Answer

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

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.32 seconds

Views:

114