Question

Calculus: How do you use trigonometric substitution on this?

Answer 1

\[\int\limits\!\frac{x^9+x^4}{(x^5-5)^{10}}dx\]

Answer 2

i would substitute \(u = x^5-5\)

Answer 3

Yes, I have split it to \[\int\limits\!\frac{x^4}{(x^5-5)^9}+\int\limits\!\frac{6x^4}{(x^5-5)^{10}}\]

Answer 4

that looks nice but partial fractions is not really needed here

Answer 5

forgot the dx there

Answer 6

direct u substitution will do

Answer 7

ah, I couldn't do a u sub on this. Care to lead the way?

Answer 8

ah, and I didn't use partial fraction. Just manipulated it a bit. I think I can handle the rest actually. A medal for your efforts my friend

Answer 9

\[\int\limits\!\frac{x^9+x^4}{(x^5-5)^{10}}dx = \int\limits\!\frac{x^4(x^5+1)}{(x^5-5)^{10}}dx\]

substitute \(u=x^5-5 \implies \frac{du}{5} = x^4dx\), the integral becomes
\[\frac{1}{5}\int\frac{u+6}{u^{10}}\,du\]

Answer 10

question. how did x^5-1 become u+6?