Integration by parts will produce a reduction formula.
i find several ways to solve this:
1)you can expand cos^5x in terms of multiples of x using euler's formula considering y=cosx+isinx and 1/y=cosx-isinx.
2)you can use a reduction formula as TKHunny said and
3)the way 'prove it' did it
4)and lastly u can perhaps use some property of definite integrals:
integration(0,pi/2)f(x)=integration(0,pi/2)f(pi/2-x).
i have not tested any of these methods though but i think they are all very feasible.