1. Integral

Hello, some of you may remember me from the previous question.

$
\int^{a}_{0} f(x)dx = \int^{a}_{0}f(a-x)dx
$

Hence evaluate

$
\int^{\frac{\pi}{2}}_{0} xsin^2xcos^2xdx
$

After messing about with it I can never seem to get it into a nicer form... thanks any help would be appreciated

2. You could try rewriting your integral.

$xsin^{2}xcos^{2}x=\frac{x^{2}}{8}-\frac{x^{2}cos(4x)}{8}$

3. Originally Posted by slevvio
Hello, some of you may remember me from the previous question.

$
\int^{a}_{0} f(x)dx = \int^{a}_{0}f(a-x)dx
$

Hence evaluate

$
\int^{\frac{\pi}{2}}_{0} xsin^2xcos^2xdx
$

After messing about with it I can never seem to get it into a nicer form... thanks any help would be appreciated
Do you just want to evaluate the integral?

Try substituting $1-cos^2(x)=sin^2(x)$ to get:

$
\int^{\frac{\pi}{2}}_{0} x(1-cos^4x)dx
$

I then suggest using tabular integration to solve that.

4. Originally Posted by slevvio
evaluate

$
\int^{\frac{\pi}{2}}_{0} xsin^2xcos^2xdx
$
Flip it around and substitute $u = \frac{\pi }
{2} - x,$
$\varphi$ becomes (say the integral):

$\varphi = \int_0^{\pi /2} {\bigg( {\frac{\pi }
{2} - x} \bigg)\sin ^2 x\cos ^2 x\,dx} .$

Hence $\varphi = \frac{\pi }
{2}\int_0^{\pi /2} {\sin ^2 x\cos ^2 x\,dx} - \int_0^{\pi /2} {x\sin ^2 x\cos ^2 x\,dx} .$

The rest follows easily, the remaining integral can be computed without problems.