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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

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1 Verified Answer | Published on 17th 09, 2020

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