$\int_0^{\pi 6} \cos ^4 3 heta \cdot \sin ^2 6 heta d heta$ equals to

Question

$\int_0^{\pi 6} \cos ^4 3 	heta \cdot \sin ^2 6 	heta d 	heta$ equals to

Aicharya · Accepted Answer

Let $I=\int_0^{\pi / 6} \cos ^4 3 	heta \sin ^2 6 	heta d 	heta$ Put $3 	heta=t \Rightarrow d 	heta=\frac{d t}{3}$ $ \begin{aligned} 	herefore \quad I & =\frac{1}{3} \int_0^{\pi / 2} \cos ^4 t \sin ^2 2 t d t \ & =\frac{1}{3} \int_0^{\pi / 2} \cos ^4 t 	imes(2 \sin t \cos t)^2 d t \ & =\frac{4}{3} \int_0^\pi \cos ^6 t \sin ^2 t d t \ & =\frac{4}{3}\left[\frac{5 \cdot 3 \cdot 1}{8 \cdot 6 \cdot 4 \cdot 2} 	imes \frac{\pi}{2}ight] \end{aligned} $ [use gamma rule of integration] $ =\frac{5 \pi}{192} $

$\int_0^{\pi 6} \cos ^4 3 \theta \cdot \sin ^2 6 \theta d \theta$ equals to

Solution