$\int_{0}^{\pi/2}\frac{dx}{1 + an^{3}x}$is equal to

Question

$\int_{0}^{\pi/2}\frac{dx}{1 + 	an^{3}x}$is equal to

Dr. Abubacker · Accepted Answer

(c)
Let $I = \int_{0}^{\pi/2}{\frac{dx}{1 + 	an^{3}x} = \int_{0}^{\pi/2}{\frac{\cos^{3}x}{\sin^{3}{x + \cos^{3}x}}dx}\ \ }$…(i)
$\Rightarrow I = \int_{0}^{\pi/2}{\frac{\cos^{3}\left( \frac{\pi}{2} - x ight)}{\sin^{3}{\left( \frac{\pi}{2} - x ight) + \cos^{3}\left( \frac{\pi}{2} - x ight)}}dx}$
$I = \int_{0}^{\pi/2}{\frac{\sin^{3}x}{\sin^{3}{x + \cos^{3}x}}dx\ }$...(ii)
On adding Eqs.(i) and (ii), we get
$2I = \int_{0}^{\pi/2}\frac{\sin^{3}{x + \cos^{3}x}}{\sin^{3}{x + \cos^{3}x}}dx$
$= \int_{0}^{\pi/2}{1dx = \frac{\pi}{2}}$
$\Rightarrow I = \frac{\pi}{4}$

Q. $\int_{0}^{\pi/2}\frac{dx}{1 + \tan^{3}x}$is equal to

Explanation

Reviewed by Dr. Abubacker

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