Question

\[\int x \sin x \cos x\ dx\]

 

Answer 1

\[\int x\sin x \cdot \text{ cos x dx }\]
\[ = \frac{1}{2}\int x\left( 2 \sin x \cos x \right) dx\]
\[ = \frac{1}{2}\int x_{} \cdot \sin \left( 2x \right)_{} dx\]
\[\text{Taking x as the first function and sin 2x as the second function} . \]
\[ = \frac{1}{2}\left[ x\int\text{ sin 2x dx } - \int\left\{ \frac{d}{dx}\left( x \right)\int\text{ sin 2x dx } \right\}dx \right]\]
\[ = \frac{1}{2}\left[ x \times \frac{- \text{ cos }\left( 2x \right)}{2} - \int1 \cdot \left( \frac{- \cos 2x}{2} \right)dx \right]\]
\[ = \frac{1}{2}\left[ \frac{- x \text{ cos
}\left( 2x \right)}{2} + \frac{\text{ sin } \left( 2x \right)}{4} \right] + C\]
\[ = \frac{- x \text{ cos } \left( 2x \right)}{4} + \frac{\text{ sin }\left( 2x \right)}{8} + C\]