Question

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

Options

•  x^sin x + C

•  x^sin x cos x + C

• \[\frac{\left( x^{\sin x} \right)^2}{2} + C\]

• none of these

Answer 1

 x^sin x + C 

\[\text{ Let I } = \int x^{\sin x} \left( \frac{\sin x}{x} + \cos x \cdot \log x \right)dx\]
\[\text{ Putting  x}^{\sin x} = t\]
\[ \Rightarrow \ln \left( x \right)^{\sin x} = \ln t\]
\[ \Rightarrow \sin x \cdot \ln x = \ln t\]
\[ \Rightarrow \left( \sin x \times \frac{1}{x} + \cos x \ln x \right)dx = \frac{1}{t}dt\]
\[ \therefore I = \int t \cdot \frac{dt}{t}\]
\[ = t + C\]
\[ = x^{\sin x} + C .............\left( \because t = x^{\sin x} \right)\]