Question

The value of $\int x^{\sin x}(\cos x\ln x+\frac{1}{x}\sin x)dx$ is
(where $C$ is constant of integration)

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

Answer 1

The correct option is A $x^{\sin x}+C$

Let $I=\int x^{\sin x}(\cos x\ln x+\frac{1}{x}\sin x)dx$
put $x^{\sin x}=t$
$\Rightarrow x^{\sin x}\cdot \frac{d}{dx}\left(\sin x\ln x\right)=dt$
$\Rightarrow x^{\sin x}\cdot (\cos x\ln x+\frac{\sin x}{x})dx=dt$

Clearly $I=\int \frac{d}{dx}\left(x^{\sin x}\right)dx$

$\Rightarrow I=x^{\sin x}+C$