Mathematics

# Solve  $\int _ { 0 } ^ { 100 } \left[ \tan ^ { - 1 } ( x ) \right] d x$

##### SOLUTION
$I=\displaystyle\int^{100}_0\tan^{-1}xdx$
Using integration by parts,
$\displaystyle\int udv=uv-\displaystyle vdu$
Let $u=\tan^{-1}x, dv=dx$
$du=\dfrac{1}{1+x^2}, v=x$
$I=x\tan^{-1}x-\displaystyle\int \dfrac{x dx}{x^2+1}$
$u_1=x^2+1$
$\Rightarrow \dfrac{du_1}{dx}=2x$
$I=x\tan^{-1}x-\dfrac{1}{2}\displaystyle\int \dfrac{du_1}{u_1}$
$=x\tan^{-1}x-\dfrac{1}{2}ln(u_1)+c$
$=\left.x\tan^{-1}x-\dfrac{1}{2}ln(x^2+1)+c\right|^{100}_0$
$=100\tan^{-1}(100)-\dfrac{1}{2}ln(10001)-0+0$
$=100\tan^{-1}(100)-\dfrac{1}{2}ln(10001)$.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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