If $I_{n} = \int_{0}^{\frac{\pi}{4}}{ an^{n}{ heta\ d heta}}$, then $I_{8} + I_{6}$ is equal to

Question

If $I_{n} = \int_{0}^{\frac{\pi}{4}}{	an^{n}{	heta\ d	heta}}$, then $I_{8} + I_{6}$ is equal to

Dr. Abubacker · Accepted Answer

(a)
$I_{8} + I_{6} = \int_{0}^{\frac{\pi}{4}}{	an^{8}{	heta\ d	heta}} + \int_{0}^{\frac{\pi}{4}}{	an^{6}{	heta\ d	heta}}$
$= \int_{0}^{\frac{\pi}{4}}{\sec^{2}	heta}	an^{6}	heta d	heta$
$= \left\lbrack \frac{	an^{7}	heta}{7} ightbrack_{0}^{\frac{\pi}{4}} = \frac{1}{7}$

Q. If $I_{n} = \int_{0}^{\frac{\pi}{4}}{\tan^{n}{\theta\ d\theta}}$, then $I_{8} + I_{6}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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