Question
$\int_0^{\pi /4} {{{\sec }^7}\theta {{\sin }^3}\theta } \,d\theta = $

Answer

c
(c) $\int_0^{\pi /4} {{{\sec }^7}\theta } .{\sin ^3}\theta \,d\theta $

$=\int_0^{\pi /4} {\frac{{{{\sin }^3}\theta }}{{{{\cos }^3}\theta }}.{{\sec }^4}\theta \,d\theta } $

$\tan \theta = t$ रखने पर,

$\int_0^1 {{t^3}(1 + {t^2})\,dt}  = $$\left| {\frac{{{t^4}}}{4} + \frac{{{t^6}}}{6}} \right|_0^1 = \frac{5}{{12}}$.

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