Question

$\int e^{x\sin x+\cos x}\left(\frac{x^4{\cos }^{3}x-x\sin x+\cos x}{x^2{\cos }^{2}x}\right)dx=$.

• A. $e^{x\sin x+\cos x}\left(x-\frac{1}{\cos x}\right)+C$
• B. $e^{x\sin x+\cos x}\left(x-\frac{1}{x\cos x}\right)+C$
• C. $e^{x\sin x+\cos x}\left(1-\frac{1}{x\cos x}\right)+C$
• D. $e^{x\sin x+\cos x}\left(1-\frac{x}{\cos x}\right)+C$

Answer 1

Answer: (B)

$e^{x\sin x+\cos x}\left(x-\frac{1}{x\cos x}\right)+C$

$\int e^{g\left(x\right)}(f^{{}^{\prime }}\left(x\right)+g^{{}^{\prime }}(x\left)f\right(x\left)\right)dx=f\left(x\right)e^{g\left(x\right)}+C$

$\Rightarrow \int e^{x\sin x+\cos x}\left(\frac{x^4{\cos }^{3}x+\cos x-x\sin x}{(x\cos x{)}^2}\right)dx=\int e^{x\sin x+\cos x}\left(\frac{d}{dx}\right(\frac{-1}{x\cos x})+x\frac{d}{dx}(x\sin x+cosx)+1-1)dx$

$\Rightarrow \int e^{x\sin x+\cos x}\left(\frac{d}{dx}\right(x-\frac{1}{x\cos x})+(x-\frac{1}{x\cos x}\left)\frac{d}{dx}\right(x\sin x+\cos x\left)\right)dx$

$\Rightarrow \int e^{x\sin x+\cos x}\left(\frac{x^4{\cos }^{3}x+\cos x-x\sin x}{(x\cos x{)}^2}\right)dx=(x-\frac{1}{x\cos x})e^{x\sin x+\cos x}+C$

So, Option B is the answer.