(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

should be my last question for at least the next few days...here goes...

Prove that, for even powers of sine,

[tex]\int^{\frac{\pi}{2}}_{0}sin^{2n}x dx = \frac{2\cdot4\cdot6\cdot...\cdot(2n - 1)}{2\cdot4\cdot6\cdot...\cdot2n}\cdot\frac{\pi}{2}[/tex]

2. Relevant equations

[tex]

uv - \int v du = \int u dvdx

[/tex]

3. The attempt at a solution

let [itex]u = sin^{2n-1}x[/itex] and [itex]dv = sin x dx[/itex]

so [itex]du = (2n-1)(sin^{2n-2}x)(cos x)dx[/itex] and [itex]v = -cos x[/itex]

and we get:

[tex]\int sin^{2n}x dx = (-cos x)(sin^{2n-1}x) + (2n-1)\int (cos^{2}x)(sin^{2n-1}x)[/tex]

and i used integration by parts again

let [itex]u = sin^{2n-2}x[/itex] and [itex]dv = cos^{2}x dx[/itex]

so [itex]du = (2n-2)(sin^{2n-3}x)(cos x)dx[/itex] and [itex]v = \frac{1}{2}((sin x)(cos x) + x)[/itex]

then we get:

[tex]\int sin^{2n}x dx = (-cos x)(sin^{2n-1}x) + \frac{2n-1}{2}(sin^{2n-2}(sinx cosx + x)) - (2n-2)\int ((sinx cosx + x)(sin^{2n-3}x)(cos x))dx[/tex]

now i've realized i'm just pointlessly integrating by parts over and over...it's just getting harder and harder (and more difficult to put here on PF!)

If someone could guide me in the right direction for proving this formula, I would appreciate it. Thank you so much in advance!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Proof of reduction formula for a definite integral of an even powered sine function

**Physics Forums | Science Articles, Homework Help, Discussion**