Should Integration by Parts Be Used on Functions Like \( x \cdot y(x) \)?

[Fluidman117]

Homework Statement

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.

Homework Equations

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

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

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

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

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

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?

 

[Jufro]

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

 

[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?

 

[pasmith]

But can someone give a good example why and when is it necessary to turn to the integration by parts technique?

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