Need Help with Trig Integrals

Out of a number of trig integrals I have been working on, there are a few I'm having trouble with. I have the answers for them but am unsure of how to proceed to reach that point.

Integral of (tan(x/2)sec(x/2))^3 dx

Answer: (2/5)sec^5(x/2) - (2/3)sec^3(x/2) + c

Integral of csc^5x dx

Answer: (-1/4)cotxcsc^3x - (3/8)cotxcscx + (3/8) ln(cscx-cotx) + c

Integral of cos(3x)cos(5x) dx

Answer: (1/4)sin(2x) + (1/16)sin(8x) + c

The final one is rather tricky.

I am not sure whether it reads:

Integral from 0 to pi/2 of sin(3x/2)sin(x/2) dx

or

Integral from 0 to pi/2 of sin^(3/2)(x)sin^(1/2)(x) dx

Either way the answer given is: (1/2)

Thanks in advance for any help given