Mathematics

Evaluate: $\displaystyle \int_{0}^{\frac{\pi}{2}} \dfrac{\sin x \cdot \cos x}{1+\sin^4 x}\cdot dx$.

SOLUTION
$\displaystyle \overset{\pi/2}{\underset{0}{\int}} \frac{\sin x \cdot \cos x}{1 + \sin^4 x} \cdot dx$.
Let $\sin^2 n = t$
$2 \sin x \,\cos x \,dx = dt$
When $x = 0, t = 0$

$x = \dfrac{\pi}{2}, t = 1$

$= \dfrac{1}{2} \displaystyle \overset{dt/2}{\underset{1 + t^2}{\int}} \frac{dt}{1 + t^2}$

$= \dfrac{1}{2} \displaystyle \overset{1}{\underset{0}{\int}} \frac{dt}{1 + t^2}$

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

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

$= \dfrac{\pi}{8}$

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Subjective Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

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