Here is #7

$\displaystyle \int_0^af(a-x)dx$ let u= a-x $\displaystyle du=-dx$

so

$\displaystyle \int_0^af(a-x)dx=\int_a^0f(u)(-du)=\int_0^af(u)du$

Now for the fun Part

$\displaystyle \int\frac{\sin^n(x)}{\cos^n(x)+\sin^n(x)}dx=\int\f rac{\sin^n(\pi/2-x)}{\cos^n(\pi/2-x)+\sin^n(\pi/2-x)}dx$

but

$\displaystyle \sin(\pi/2-x)=\cos(x)$ and $\displaystyle \cos(\pi/2-x)=sin(x)$

$\displaystyle \int\frac{\sin^n(x)}{\cos^n(x)+\sin^n(x)}dx=\int\f rac{\cos^n(x)}{\cos^n(x)+\sin^n(x)}dx= L$ Then

$\displaystyle \int\frac{\sin^n(x)}{\cos^n(x)+\sin^n(x)}dx+\int\f rac{\cos^n(x)}{\cos^n(x)+\sin^n(x)}dx =2L $

summing up the left side and reducing we get...

$\displaystyle \int_0^{\pi/2}dx=2L$ or

$\displaystyle \frac{\pi}{2}=2L \iff \frac{\pi}{4}=L$

YEAH!!!