Integration by parts expression

[natashajane]

Use integration by parts to express:

I (n) = ∫(sin)^n (x) dx in terms of I (n-2)


Let u = sinn-1 x
du = (n-1)sinn-2 x cos x dx
v = - cos x
dv = sinx dx

so integration by parts give:

∫〖sin〗^n x dx= -cos⁡〖x 〖sin〗^(n-1) x+(n-1) ∫〖sin〗^(n-2) 〗 x cos^2⁡〖x dx〗

Since cos2x = 1 – sin2x, we have:

∫〖sin〗^n x dx= -cos⁡〖x 〖sin〗^(n-1) x+(n-1) ∫〖sin〗^(n-2) 〗 x dx-(n-1)∫〖sin〗^n x dx

We solve this equation by taking the last term on the right side to the left side:

∫〖sin〗^n x dx= -cos⁡〖x 〖sin〗^(n-1) x+(n-1)∫〖〖sin〗^(n-2) x dx〗〗

Is this right?

 

[Schrodinger's Dog]

Differentiate your answer using the same terms, see what you end up with.