When Should You Use Linearity, Substitution, or Partial Integration?

[TKay]

Hello, Forum!


I just registered after seeing you actually help people understand their problems. That's great.

We have (or should have) learned about linearity, substitution and partial integration. However, I don't know when to use which! Could someone also give me a bit of an expanation on this? :(

I have to solve an integral:
x*e^(-3x) dx

My train of thought: I have almost got 2 'basis integrals': x dx and e^x dx. I probably need to substitute to get them to the basic form. But how!
As you see I'm pretty clueless, but what I came up with was:
u = -3x --> u'= -3
v' = x --> v = (x²)/2
However, this leads nowhere. I don't know what to do!

According to derive, the solution is supposed to be:

     1           -3x  ⎛    x            1       ⎞ 
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ - e           ⎜⎯⎯⎯⎯⎯⎯⎯⎯⎯ + ⎯⎯⎯⎯⎯⎯ ⎟
          2            ⎜ 3·LN(e)              2⎟
 9·LN(e)             ⎝               9·LN(e) ⎠

I sincerely hope someone will be able to show me the light!
Thanks in advance.

PS: Our teacher is really bad at teaching!

 

[Kurdt]

Try u = x and v' = e^(-3x)

In integration by parts what you're looking to do is reduce any x factors to a constant and thus you set those equal to u. This reduces the right hand side integral to a single function which should be easy to deal with.

If the x factors are higher powers then apply the integration by parts method until the x reduces to a constant or you can come up with a reduction formula.

The substitution method is a little intuitive because you're looking for something that is a derivative of something else in the function. Just keep practising some substitution questions and you will soon start to spot them fairly easily.

For example:

$$\int xe^{x^2} dx$$

You can spot that x is almost the derivative of x². So we use the following substitution:

$$u=x^2$$ therefore $$\frac{du}{dx}=2x \Rightarrow xdx=\frac{1}{2}du$$

$$\frac{1}{2} \int e^u du = \frac{1}{2} e^{x^2} + c$$

 

[Joza]

The LIATE rule helps you identify which one to use as u.
Order of priority is:

Logarithms, Inverse Trigonometric, Algebraic, Trigonometric, Exponential