# Homework Help: Integration by parts

1. Sep 5, 2009

### clairez93

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

$$\int\frac{t^{2}}{\sqrt{2+3t}}$$

Use integration by parts to verify the formula:
$$\int x^{n} sin x dx = -x^{n} cos x + n\int x^{n-1} cos x dx$$

2. Relevant equations

3. The attempt at a solution

For the first one, I attached the picture of my work on paper, as it would take me forever to type out in latex code, I think. For the second one:

$$u = sin x$$
$$du = cos x$$
$$dV = x^{n}$$
$$V = \frac{x^{n+1}}{n+1}$$

$$\int x^{n} sin x dx = -x^{n} cos x + n\int x^{n-1} cos x dx$$ =
$$sin x (\frac{x^{n+1}}{n+1}) - \int \frac{x^{n+1}}{n+1} cos x dx$$

That doesn't really look like the formula to me. Am I supposed to use an identity of some sorts?

File size:
15.6 KB
Views:
76
2. Sep 5, 2009

### mathie.girl

I can't see the work for the first one, so I can't tell if that's right or not.

For the second one, you differentiated $$\sin{x}$$ to get $$\cos{x}$$and integrated $$x^n$$ to get $$\frac{x^{n+1}}{n+1}$$, is that right? However, since the formula you're supposed to end up with has an $$x^{n-1}$$, I would differentiate the $$x^n$$ and integrate the $$\sin{x}$$ and see what you get.

3. Sep 5, 2009

### n!kofeyn

mathie.girl is right. A good thing to remember when doing integration by parts is that you let u be the term such that when you differentiate it, du is "simpler" than u. For example, you let u=sinx so that du=cosx. That really doesn't do any simplifying. That means you'll let dv be the term that doesn't really simplify when taking the derivative.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook