Question

Solve:

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

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

Answer 1

Answer: (A)

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

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

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

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

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

$=x{e}^{\sin x}-\int e^{\sin x}dx-e^{\sin x}\sec x+\int e^{sinx}\cos x\sec x+c$

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

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