Question

${\int }_0^{\frac{\pi }{4}}se{c}^7\theta si{n}^3\theta d\theta =$

• A. $\frac{1}{12}$
• B. $\frac{3}{12}$
• C. $\frac{5}{12}$
• D. $Noneofthese$

Answer 1

The correct option is C $\frac{5}{12}$

${\int }_0^{\frac{\pi }{4}}se{c}^7\theta .si{n}^3\theta d\theta ={\int }_0^{\frac{\pi }{4}}\frac{si{n}^3\theta }{co{s}^3\theta }.se{c}^4\theta d\theta$
Putting tan $\theta$ = t, it reduces to
${\int }_0^1{t}^3(1+t^2)dt={|\frac{t^t}{4}+\frac{t^6}{6}|}_0^1=\frac{5}{12}$