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

[MMM]

Homework Statement

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.

Homework Equations

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.

 

[Quesadilla]

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.

 

[MMM]

Thanks for the help. I understand it now!

 

[Quesadilla]

To practice and since you knew how to integrate $x/\sqrt{1 - x^4}$, how would you tackle
\begin{equation*}
\frac{ax}{\sqrt{b - cx^4}}?
\end{equation*}