# Homework Help: Trigonometric reduction

1. Apr 19, 2007

### JamesGregory

1. The problem statement, all variables and given/known data

Prove the following trigonometric reduction using integration by parts:
$$\int \sin^n x dx = - \frac{\ \sin^{n-1} x \cos x}{n} + \frac{\ n-1}{n} \int \sin^{n-2} x dx$$

2. The attempt at a solution

I tried using integration by parts by breaking up sin^n x into sin^(n-2) x sin^2 x and sin^(n-1) x sin x but couldn't seem to get either to work. If someone could just provide a useful link as opposed to typing the whole proof, that would be appreciated.

Last edited: Apr 19, 2007
2. Apr 19, 2007

### HallsofIvy

Did you consider induction on n?
It's obvious that this is true when n= 1.

Assume the statement is true for sink x and use integration by parts on
[tex]\int sin^{k+1} xdx= \int sin x sin^k x dx[/itex]
with u= cos x and dv= sink dx.

3. Apr 19, 2007

### JamesGregory

This is high school ap calculus ab, and I don't think we've gotten to induction, but I do think I see what you're getting at. but shouldn't u and dv be parts of the original integrand or am I missing something obvious?

4. Apr 19, 2007

### Gib Z

Ahh yes I think Halls just typoed, but I he should mean u = sin x, and dv = sin^k x.

5. Apr 20, 2007

### dextercioby

In my hs, we got induction method 2 yrs before integral calculus...

6. Apr 20, 2007

### HallsofIvy

This is a high school class, you haven't done proof by induction, but you have done integration? Am I the only one who thinks that is backwards?

7. Apr 20, 2007

### Gib Z

I was about to say that in my post as well, but seeing as Australia does about everything backwards maybe I didn't know what i was talking about. Though mathwonk always like to say how some of his Calculus students don't know basic algebra and how it appalls him. Something's seriously wrong with that education system.