Write: #cos^4x = cos^3x*cosx# and integrate by parts:
#int cos^4xdx = int cos^3x cosx dx = int cos^3x d(sinx)#
#int cos^4xdx = sinxcos^3x - int sinx d(cos^3x)#
#int cos^4xdx = sinxcos^3x + 3int sin^2x cos^2xdx#
Now use the identity:
#sin^2x = 1-cos^2x#
#int cos^4xdx = sinxcos^3x + 3int (1-cos^2x) cos^2xdx#
#int cos^4xdx = sinxcos^3x + 3int cos^2x dx -3int cos^4xdx#
We have now the same integral on both sides and we can solve for it:
#4 int cos^4xdx = sinxcos^3x + 3int cos^2x dx #
# int cos^4xdx = (sinxcos^3x)/4 + 3/4int cos^2x dx #
Using the same process:
#int cos^2x dx = int cosxd(sinx) = cosxsinx + int sin^2xdx#
#int cos^2x dx = int cosxd(sinx) = cosxsinx + int (1-cos^2x)dx#
#int cos^2x dx = int cosxd(sinx) = cosxsinx + x - int cos^2xdx#
#int cos^2x dx = (cosxsinx)/2 + x/2#
Substituting in the expression above:
# int cos^4xdx = (sinxcos^3x)/4 + 3/8(cosxsinx) + 3/8x#
