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
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