While browsing through some notes on integral substitution i found a neat little problem that i would like very much to check my understanding of with you. I am to suggest an integral that is reduced to a rational function integral when this substitution is used:
$a)$ $t=\sin x$
$b)$ $t=\sqrt[6] {x+5}$
$c$ $\sqrt{1-9x^2}=-1+xt$
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) $\int \sqrt {ax^2+bx+1} \, dx$ where $a,b$ can be any number. Care to share your thoughts on this? Perhaps some suggestions of your own or corrections on mine?
Thanks, :)