## mpj4 one year ago Calculus: How do you use trigonometric substitution on this?

1. mpj4

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

2. rational

i would substitute $$u = x^5-5$$

3. mpj4

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

4. rational

that looks nice but partial fractions is not really needed here

5. mpj4

forgot the dx there

6. rational

direct u substitution will do

7. mpj4

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

8. mpj4

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

9. rational

$\int\limits\!\frac{x^9+x^4}{(x^5-5)^{10}}dx = \int\limits\!\frac{x^4(x^5+1)}{(x^5-5)^{10}}dx$ substitute $$u=x^5-5 \implies \frac{du}{5} = x^4dx$$, the integral becomes $\frac{1}{5}\int\frac{u+6}{u^{10}}\,du$

10. mpj4

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