Question

Let $\int(\sin 3 \theta+\sin \theta) \cos \theta \cdot e^{\sin \theta} \cdot d \theta=\left(A \cdot \sin ^3 \theta+B \cdot \cos ^2 \theta+C \cdot \sin \theta+D \cdot \cos \theta+E\right) \cdot e^{\sin \theta}+K$
where $K$ is constant of integration. Find the value of $\frac{B}{A}+\frac{D}{E}$.

Answer 1

Let $I=\int 2 \cdot \sin 2 \theta \cdot \cos ^2 \theta \cdot e^{\sin \theta} d \theta=\int 4 \sin \theta \cdot \cos ^3 \theta \cdot e^{\sin \theta} \cdot d \theta$
Put $\sin \theta=t$, so
$I =\int 4 t \left( l - t ^2\right) e ^{ t } dt =4 \int_{ I }^{ t } \cdot e ^{ t }-4 \int t ^3 \cdot e ^{ t } dt $
$\therefore I =4 I _1-4 I _2 $
$\text { We get, } I =-4 \sin ^3 \theta-12 \cos ^2 \theta-20 \sin \theta+32 $
$\therefore A =-4, B =-12, C =-20, D =0, E =32 $