Question

Evaluate $\int \frac{e^{\sin x}(x{\cos }^{3}x-\sin x)}{{\cos }^{2}x}dx$

• A. $e^{\sin x}(x-\sec x)+c$
• B. $e^{\sin x}(x+\sec x)+c$
• C. $e^{\sin x}(x\cos x+\sin x)+c$
• D. $e^{\sin x}(x\cos x-\sin x)+c$

Answer 1

The correct option is A $e^{\sin x}(x-\sec x)+c$

$\int e^{\sin x}(\frac{x{\cos }^{3}x}{{\cos }^{2}x}-\frac{\sin x}{{\cos }^{2}x})dx$

$=\int e^{\sin x}(x\cos x-\tan x\sec x)dx.$

$=\int e^{\sin x}\left(x\cos x\right)dx-\int e^{\sin x}\left(\tan x\sec x\right)dx$

$=x\int e^{\sin x}\cos xdx-\int 1e^{\sin x}dx.-[e^{\sin x}\int \tan x\sec x-\int e^{\sin x}\cos x\sec xdx]$ .... integration by parts

$=x\int e^{\sin x}\cos xdx-\int e^{\sin x}dx.-e^{\sin x}\int \tan x\sec x+\int e^{\sin x}dx$

$=x{e}^{\sin x}-e^{\sin x}\sec x+c$

$=e^{\sin x}(x-\sec x)+c$