Mathematics

Evaluate $$\displaystyle\int_{0}^{\pi/2}\dfrac{\cos x}{1+\sin^{2}x}dx$$


SOLUTION
$$\displaystyle\int_{0}^{\frac{\pi}{2}}{\dfrac{\cos{x}}{1+{\sin}^{2}{x}}dx}$$

Let $$t=\sin{x}\Rightarrow\,dt=\cos{x}dx$$

As $$x\rightarrow\,0\Rightarrow\,t\rightarrow\,0$$

$$x\rightarrow\,\dfrac{\pi}{2}\Rightarrow\,t\rightarrow\,1$$

$$\displaystyle\int_{0}^{\frac{\pi}{2}}{\dfrac{\cos{x}}{1+{\sin}^{2}{x}}dx}$$

$$=\displaystyle\int_{0}^{1}{\dfrac{dt}{1+{t}^{2}}}$$

$$=\left[{\tan}^{-1}{t}\right]_{0}^{1}$$

$$={\tan}^{-1}{1}-{\tan}^{-1}{0}$$

$$=\dfrac{\pi}{4}-0$$

$$=\dfrac{\pi}{4}$$
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Subjective Medium Published on 17th 09, 2020
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