#### Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions Question 5 Maths Textbook Solution.

$x^{\sin x}+C$

Given:

$\int x^{\sin x}\left(\frac{\sin x}{x}+\cos x \cdot \log x\right) d x$

Hint:

You must know about the derivation of $x^{\sin x}$

Explanation:

Let $\mathrm{I}=\int x^{\sin x}\left(\frac{\sin x}{x}+\cos x \cdot \log x\right)$

$=\int t \cdot \frac{d t}{t}$                                                                                     $\text { [ Put } x^{\sin x}=t, \text { Taking log of both sides }$

$=\int 1 d t$                                                                                          $\sin x \cdot \log x=\log t$

$=t+C$                                                                                           Diff.  w. r. t.  t,

$=x^{\sin x}+C$                                                                           $\left.\left(\log x \cos x+\frac{\sin x}{x}\right) \frac{d x}{d t}=\frac{1}{t} \Rightarrow\left(\log x \cos x+\frac{\sin x}{x}\right) d x=\frac{d t}{t}\right]$