Question

$\int x^{\sin x}\left(\frac{\sin x}{x}+\cos x.\log x\right)dx\mathrm{is}\mathrm{equal}\mathrm{to}$

(a) x^{sin x} + C

(b) x^{sin x} cos x + C

(c) $\frac{{\left(x^{\sin x}\right)}^2}{2}+C$

(d) none of these

Answer 1

(a) x^{sin x} + C

$$\begin{gathered} \mathrm{Let}I=\int x^{\sin x}\left(\frac{\sin x}{x}+\cos x·\log x\right)dx \\ \mathrm{Putting}{x}^{\sin x}=t \\ \Rightarrow \ln {\left(x\right)}^{\sin x}=\ln t \\ \Rightarrow \sin x·\ln x=\ln t \\ \Rightarrow \left(\sin x\times \frac{1}{x}+\cos x\ln x\right)dx=\frac{1}{t}dt \\ \therefore I=\int t·\frac{dt}{t} \\ =t+\mathrm{C} \\ =x^{\sin x}+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }\mathit{}t\mathit{=}\mathit{}{x}^{\sin \mathit{}x}\right) \end{gathered}$$