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Step-by-step Solution

Integrate (x^2+4)^0.5

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Answer

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

Step-by-step explanation

Problem to solve:

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

Solve the integral $\int\sqrt{x^2+4}dx$ by trigonometric substitution using the substitution

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

Substituting in the original integral, we get

$\int2\sqrt{4\tan\left(\theta\right)^2+4}\sec\left(\theta\right)^2d\theta$

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Answer

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

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$\int\sqrt{x^2+4}dx$

Main topic:

Integration by trigonometric substitution

Used formulas:

9. See formulas

Time to solve it:

~ 1.29 seconds