Integrate by parts:
#int cos^6xdx = int cos^5x cosx dx = int cos^5x d(sinx)#
#int cos^6xdx = sinxcos^5x + 5 int cos^4x sin^2x dx#
#int cos^6xdx = sinxcos^5x + 5 int cos^4x (1 - cos^2x) dx#
#int cos^6xdx = sinxcos^5x + 5 int cos^4x - 5int cos^6x dx#
As the integral is on both sides we can solve for it:
#6 int cos^6xdx = sinxcos^5x + 5 int cos^4xdx #
#int cos^6xdx = (sinxcos^5x )/6+ 5/6 int cos^4xdx #
We can use the same method to reduce the degree of #cosx# again:
#int cos^4xdx = int cos^3x d(sinx) #
#int cos^4xdx = sinxcos^3x + 4 int cos^2x sin^2x dx #
#int cos^4xdx = sinxcos^3x + 4 int cos^2xdx -4 int cos^4xdx #
#int cos^4xdx = (sinxcos^3x)/5 + 4/5 int cos^2xdx #
And again:
#int cos^2xdx = int cosx d(sinx) #
#int cos^2xdx = sinxcosx + int sin^2x dx #
#int cos^2xdx = sinxcosx + int dx - int cos^2xdx #
#int cos^2xdx = (sinxcosx)/2 + 1/2x +C #
Putting it all together we have:
#int cos^6xdx = (sinxcos^5x )/6+ (sinxcos^3x)/6 + 2/5 (sinxcosx) + 2/5x +C#
