Question

$\int x\sin \frac{x}{2}=?$

Answer 1

Sol:

$\int x\sin \frac{x}{2}dx$

Let

$I=\int x\sin \frac{x}{2}dx$

We know that

$I=\int \Pi dx=I\int \Pi dx-\int \left(\frac{dI}{dx}\int \Pi dx\right)dx+C$

Now, using $LAT\epsilon$

$I=\int x\sin \frac{x}{2}dx=x\int \sin \frac{x}{2}dx-\left[\int \left(\frac{dx}{dx}\int \sin \frac{x}{2}dx\right)dx\right]+C$

$=x\frac{[-\cos \frac{x}{2}]}{\frac{1}{2}}-\int \frac{(-\cos \frac{x}{2})}{\frac{1}{2}}dx+C$

$=-2x\cos \frac{x}{2}+\int 2\cos \frac{x}{2}dx+C$

$=2x\cos \frac{x}{2}+2\int \cos \frac{x}{2}dx+C$

$=-2cos\frac{x}{2}+2\left[\frac{\frac{\sin x}{2}}{\frac{1}{2}}\right]+C$

$=-cos\frac{x}{2}+4\sin \frac{x}{2}+C$

Hence, this is the answer.