Use #intcos^n(x)dx= 1/(n)cos^(n-1)(x)sin(x)+(n-1)/nintcos^(n-2)(x)dx# where n = 6:
#intcos^6(x)dx= 1/(6)cos^5(x)sin(x)+5/6intcos^4(x)dx#
Use #intcos^n(x)dx= 1/(n)cos^(n-1)(x)sin(x)+(n-1)/nintcos^(n-2)(x)dx# where n = 4:
#intcos^6(x)dx= 1/(6)cos^5(x)sin(x)+5/6[1/4cos^3(x)sin(x)+3/4intcos^2(x)dx]#
Use #intcos^n(x)dx= 1/(n)cos^(n-1)(x)sin(x)+(n-1)/nintcos^(n-2)(x)dx# where n = 2:
#intcos^6(x)dx= 1/(6)cos^5(x)sin(x)+5/6[1/4cos^3(x)sin(x)+3/4{1/2cos(x)sin(x)+1/2x}]+C#
Simplify:
#intcos^6(x)dx= 1/(6)cos^5(x)sin(x)+5/24cos^3(x)sin(x)+15/24{1/2cos(x)sin(x)+1/2x}+C#
#intcos^6(x)dx= 1/(6)cos^5(x)sin(x)+5/24cos^3(x)sin(x)+15/48cos(x)sin(x)+15/48x+C#
Evaluate at the limits:
#int_0^(pi/2)cos^6(x)dx= [1/(6)cos^5(x)sin(x)+5/24cos^3(x)sin(x)+15/48cos(x)sin(x)+15/48x]_0^(pi/2)#
#int_0^(pi/2)cos^6(x)dx= (5pi)/32#