integrate ..u substitution

[bimbo36]

 

what do i do here ?

[Schrödinger's hat]

 

what do i do here ?

 

The most common situation for u substitution is to use the chain rule in reverse to turn something you can't integrate into something you can.

 

What we want to do is manipulate our equation into something that looks like this:

 

[math] \int f(u(x)) \frac{du}{dx} dx [/math]

 

So a function of a second function, with the derivative of the second function multiplying the whole lot.

 

Then we can use the chain rule to replace it with:

[math] \int f(u) du [/math]

 

A common example is:

[math] \int x e^{x^2} dx [/math]

 

First we observe that the derivative of [math] x^2[/math] is [math]2x[/math]

So [math]u=x^2[/math]

We can easily manipulate things to get a constant by multiplying by 1.

[math] \int x e^{x^2} dx = \frac{2}{2} \int x e^{x^2} dx = \frac{1}{2} \int 2x e^{x^2} dx [/math]

 

Which is:

[math] \frac{1}{2}\int e^{u} du = \frac{1}{2} e^{u} + C = \frac{1}{2} e^{x^2} + C [/math]

 

 

Hopefully this is enough that you can apply the same logic to your function. (If not, ask again and I'll elaborate further)

Edited November 25, 2011 by Schrödinger's hat

[mathematic]

 

what do i do here ?

 

u=x^3

du=3x^2dx

 

You should be able to continue.