Calc 4 question

Question

Prove that for n ≥ 2

∫ cos^(n)x dx = (1/n)cos^(n−1)x • sin x + (n − 1)/n • ∫ cos^(n−2)x dx

Answer 1

(Assuming that n is an integer ≥ 2)

 

Use integration by parts:

 

∫cosⁿx dx = ∫cos^n-1x cosx dx

 

                       Let u = cos^n-1x     dv = cosxdx

 

                           du = (n-1)cos^n-2x(-sinx)dx    v = ∫dv = sinx

 

So, ∫cosⁿx dx = uv - ∫vdu = cos^n-1x sinx - ∫sinx(n-1)cos^n-2x(-sinx)dx

 

                                       = cos^n-1x sinx + (n-1)∫cos^n-2x sin²x dx

 

                        ∫cosⁿxdx = cos^n-1x sinx + (n-1)∫cos^n-2x (1 - cos²x)dx

 

                        ∫cosⁿxdx = cos^n-1x sinx + (n-1)∫cos^n-2x dx - (n-1)∫cosⁿx dx

 

Combining the terms that involve ∫cosⁿxdx, we have n∫cosⁿ xdx = cos^n-1x sinx + (n-1)∫cos^n-2x dx 

 

Divide both sides by n to obtain the desired formula.