Question

Evaluate the following indefinite integrals.

Answer 1

Hmm looks like a straightforward U-substitution c:

Familiar with the process?

Answer 2

Yeah, I know how to solve it. Just checking my work, that's all. Doesn't hurt to check with someone else. :D

Answer 3

\[\large \int\limits\frac{2x^2+2x}{\left(x^3+3x^2+9\right)^4}dx\]

Let \(\large \color{orangered}{u=x^3+3x^2+9}\),
Taking the derivative with respect to X gives us,\[\large \frac{du}{dx}=3x^2+6x\]Moving the dx to the other side and factoring out a 3 gives us,\[\large \color{orangered}{\frac{1}{3}du=(2x^2+2x)dx}\]

Plugging in the orange pieces gives us,\[\large \int\limits\limits \frac{\frac{1}{3}du}{u^4} \quad = \quad \frac{1}{3}\cdot \left(-\frac{1}{3}\right)u^{-3}+C\]

Answer 4

Putting it back into X gives ussssssssssssss,\[\large -\frac{1}{9}\left(x^3+3x^2+9\right)^{-3}+C\]

I think... Lemme check over my work real quick :D
Did you come up with something like this?

Answer 5

Yep! Except, mines formatted differently but nonetheless the same thing. :)

Answer 6

cool c: