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Step-by-step Solution

Integrate $\sqrt{4x^2+1}$ with respect to x

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Answer

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

Step-by-step explanation

Problem to solve:

$\int\:\left(\sqrt{4x^2+1}\right)$
1

First, factor the terms inside the radical for an easier handling

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

Taking the constant out of the radical

$\int2\sqrt{x^2+\frac{1}{4}}dx$

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Answer

$\frac{1}{2}\left(\sqrt{4x^2+1}x+\frac{1}{2}\ln\left|\sqrt{4x^2+1}+2x\right|\right)+C_0$
$\int\:\left(\sqrt{4x^2+1}\right)$

Main topic:

Integration by trigonometric substitution

Used formulas:

9. See formulas

Time to solve it:

~ 1.01 seconds

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