Question

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

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

Answer 1

Answer: (A)

$x{e}^{\sin x}-e^{\sin x}.\sec x$

$I=\int [x\left(e^{\sin x}\cos x\right)-e^{\sin x}\left(\sec x\tan x\right)]dx$
Integrate by parts.
$[x{e}^{\sin x}-\int e^{\sin x}.1.dx]-[e^{\sin x}\sec x-\int e^{\sin x}\cos x\sec xdx]$
$=x{e}^{\sin x}-e^{\sin x}.\sec x$ other integrals cancel