F.5 A.maths - Simple Techniques of Integration

The technique is posted here.
1) S (sin2xcos3x) dx
product of sinx * cosy is transformed to 1/2* {sin[(x+y)/2] - sin[(x-y)/2]}
Then integrate!
2) S (sin(^2)2x) dx
change sin^2(y) into 1/2(1-cos2y)
Then integrate!
3) S (sin(^3)xcosx) dx
OH! There is single cosx!!
Let u = sinx
du = cosxdu
then integrate in terms of u.

4) S (sin(^4)xcos(^4)x) dx

OH! There is no single cosx or sinx!
view it = [sinxcosx]^4 = [1/2sin2x]^4 with even index
Transform it into compound angle of cos2x, cos4x,
5) S (cos(^3)x) dx
view it = [cosx]^3 with odd index
view it = [cosx]^2*cosx
let u = sinx
du = cosxdx
Transform into u. Then integrate!

6) S (sin(^4)x) dx
it is even index!
sin(^4)x=[1/2(1-cos2x)] ^2= expand it in terms of cos2x, [cos2x]^2. Then change [cos2x]^2 in terms of 1/2[cos4x+1]
Then integrate.

1) = 1/2cosx-1/10cos5x
2) = 1/2x -1/8sin4x