Mathematics

$$\int\limits_0^{\pi /2} {\dfrac{{\sin x\cos x}}{{1 + {{\sin }^4}x}}dx = } $$


SOLUTION
Integrate $$I=\displaystyle\int_{0}^{x/2}\dfrac{\sin x\cos x}{\sin^{4}x}dx$$

Let $$\sin^{2}x=t$$ at $$x=0\Rightarrow t=0$$

Differentiating  w.r.t $$x$$ at $$x=x/2 \Rightarrow t=2$$
$$2\sin x\cos x dx=dt$$
$$\sin x\cos x dx=dt/2$$

$$\therefore I=\dfrac{1}{2}\displaystyle\int_{0}^{1}\dfrac{dt}{(\sin^{2}x)^{2}+1}$$
$$=\dfrac{1}{2}\displaystyle\int_{0}^{1}\dfrac{dt}{t^{2}+1}$$

$$=\dfrac{1}{2}[\tan^{-1}t]_{0}^{1}$$

$$=\dfrac{1}{2}\left[\tan^{-1}1-\tan^{-1}0\right]=\dfrac{1}{2}\left[\dfrac{\pi}{4}=0\right]$$

$$=\dfrac{\pi}{8}$$
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Subjective Medium Published on 17th 09, 2020
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