# A simple question on integration

1. May 12, 2014

### Fluidman117

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

I want to take an antiderivative of a function with respect to x. But in addition the function includes a term y (x) that is a function of x itself. Do I have to apply the reverse power rule also to y(x) also? The integral can be seen as an indefinite.

2. Relevant equations

$A=\int x*y(x)*dx$

3. The attempt at a solution

I think I should just apply the reverse power rule to x. So:

$A= \frac{x^{2}}{2}*y(x) + C$

2. May 12, 2014

### Ray Vickson

You can answer your own question---and that is the best way to learn! Just try it out on some examples. What do you get if you use your formula on the function $y(x) = x^2$? What happens if you use your formula on $y(x) = 1/x^2$? Are you getting correct results?

Last edited: May 12, 2014
3. May 12, 2014

### Jufro

What you are looking for is something called integration by parts.

4. May 14, 2014

### Fluidman117

Okay, then I believe I was mistaken. It seems that I also need to apply the reverse power rule to $y(x)$.
So an example:
$y(x)=2*x$

$A=\int^{2}_{1} x*y(x)*dx=\int^{2}_{1} x*2*x*dx$
$A=\left|2\frac{x^{3}}{3}\right|^{2}_{1}=4.6667$

Is this correct?

I also looked up the integration by parts and it seems that in the above example it was possible to do without the integration by parts. But can someone give a good example why and when is it necessary to turn to the integration by parts technique?

5. May 14, 2014

### pasmith

Try $\displaystyle\int_0^\pi x \sin x\,dx$ or $\displaystyle\int_0^1 x e^{-x}\,dx$.