# Homework Help: Integration by substitution

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1. Mar 12, 2017

### doktorwho

1. The problem statement, all variables and given/known data
Suggest an integral that is reduced to a rational function integral when this substitution is used:
$a)$ $t=\sin x$
$b)$ $t=\sqrt[6] {x+5}$
$c$ $\sqrt{1-9x^2}=-1+xt$
2. Relevant equations
3. The attempt at a solution

I found this to be a very interesting problem and wanted to check my results with you. For the first part i think that
a) $\int \sin x\cos x \, dx$ is a good idea couse when we introduce the given substitution we are left with $\int t \, dt$ which is a rational function, right?
For the second part i have some struggles but i think that
b) $\int \frac{\sqrt[6] {(x+5)^5}}{6\sqrt[6] {x+5}} \, dx$ would reduce to $\int t \, dt$
As for the part c) that i the second Euler substitution and the integral that is suited for it should be any integral of the form c) $\int \sqrt {ax^2+bx+1} \, dx$
How does this seem to you? Any comments? Any feedback is appreciated

2. Mar 13, 2017

### Orodruin

Staff Emeritus
The easiest way is to start with an integral of a rational function and do the substitution in the other direction ...

3. Mar 13, 2017

### doktorwho

I'm gonna try that as well, could you take a look at what i came up, to make sure i got it right or wrong?