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Step-by-step Solution

Integrate $\frac{arctan\left(x\right)}{x^2+1}$ from $0$ to $\infty $

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Answer

$\frac{1}{2}arctan\left(\infty \right)^2$

Step-by-step explanation

Problem to solve:

$\int_0^{\infty}\left(\frac{\arctan\left(x\right)}{x^2+1}\right)dx$
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Replace the integral's limit by a finite value

$\lim_{c\to\infty }\:\int_{0}^{c}\frac{arctan\left(x\right)}{x^2+1}dx$
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Solve the integral $\int_{0}^{c}\frac{arctan\left(x\right)}{x^2+1}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=arctan\left(x\right) \\ du=\frac{1}{1+x^2}dx\end{matrix}$

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Answer

$\frac{1}{2}arctan\left(\infty \right)^2$