# How do I know what to substitute(u-substitution)

1. Jul 31, 2014

### MMM

1. The problem statement, all variables and given/known data
Hello, I'm having trouble determining what I make u equal with most integrals that revolve around the inverse trig functions.

I knew what to let u equal in this problem $x/\sqrt{1-x^4}$ I let it equal $x^2$ and correctly solved the problem.

The problems I'm having trouble with, are problems like $1/(a^2 + x^2)$ the book said to substitute u = x/a and when I did that I found the correct answer. Sadly I had no idea that is what I was supposed to substitute.

Here is another problem I had no idea what to substitute $1/(a^2 + (b^2)(x^2))$ the book said to substitute u = bx/a and when I did that I got the correct answer. I just need some guidance to understand the kinda substitutions I need to make regarding integration involving inverse trig functions. Any help is greatly needed and highly appreciated, thanks.

2. Relevant equations

3. The attempt at a solution
I solved the stated problems with the book's help on what to substitute. I just had no idea on to what to substitute.

2. Jul 31, 2014

You are looking for something you know how to integrate. In this case when you see something like $1/(a^2 + b^2x^2)$, your first instinct should be that it looks kind of like $1/(1 + x^2)$ and so we want to substitute to put it on that form.

Here we note that
\begin{equation*}
\frac{1}{a^2 + b^2 x^2} = \frac{1}{a^2(1 + \frac{b^2}{a^2}x^2)}= \frac{1}{a^2} \frac{1}{1 + (\frac{b}{a}x)^2}
\end{equation*}
and the substitution suggests itself.

You see these things easier and faster as you get more and more experience.

3. Jul 31, 2014

### MMM

Thanks for the help. I understand it now!

4. Jul 31, 2014

To practice and since you knew how to integrate $x/\sqrt{1 - x^4}$, how would you tackle