Question

17. Integrate sin x* e^cosx dx
A) e^sinx + C
C) -e^cosx + C
B) e^cosx + C
D) -e^sinx+C​

Answer 1

We're asked to evaluate,

$\displaystyle\longrightarrow I=\int\sin x\cdot e^{\cos x}\ dx\quad\quad\dots(1)$

Substitute,

$\displaystyle\longrightarrow u=e^{\cos x}\quad\quad\dots(2)$

$\displaystyle\longrightarrow du=d(e^{\cos x})$

$\displaystyle\longrightarrow du=e^{\cos x}\cdot-\sin x\ dx$

$\displaystyle\longrightarrow -du=e^{\cos x}\cdot\sin x\ dx$

$\displaystyle\longrightarrow \sin x\cdot e^{\cos x}\ dx=-du$

Then (1) becomes,

$\displaystyle\longrightarrow I=\int-du$

$\displaystyle\longrightarrow I=-\int du$

$\displaystyle\longrightarrow I=-u+C$

From (2),

$\displaystyle\longrightarrow\underline{\underline{I=-e^{\cos x}+C}}$

Hence (C) is the answer.