Question

Find the indefinite integral:
Integrand of (4x^3+2)/((x^4+2x)^2) dx

Answer 1

u=(x^4+2x), du= 12x^2 dx

Answer 2

how do I simplify from here since its a fraction?

Answer 3

that'll do it!

Answer 4

well actually it is \(u=x^4+2x,du=(4x^3+2)dx\)

Answer 5

then integral becomes
\[\int \frac{du}{u^2}\] use the power rule backwards to solve

Answer 6

umm.. wouldn't du be 12x^2 since that is the derivative of (4x^3+2)?

Answer 7

so I would do the anti derivative of (du)/u^2

Answer 8

wait wait
you do not take the derivative twice, just once

Answer 9

\[u(x)=x^4+2x\]
\[\frac{du}{dx}=4x^3+2\]
\[du=(4x^3+2)dx\] like that, which as luck would have it is exactly what you have in your problem

Answer 10

so yes, you take the anti derivative of \(\frac{1}{u^2}\) which is straight forward

Answer 11

i see..

Answer 12

you get \(-\frac{1}{u}\) by the power rule backwards, then replace \(u\) by \(x^4+2x\) for your answer

Answer 13

would my answer be -(x^4+2x)^(-1)+C?

Answer 14

-1/(x^4+2 x)+constant

Answer 15

oh ok.. i move (x^4+2x) down to be the denominator.

Answer 16

yes that is one way to write it with exponential notation
easier way to understand is
\[-\frac{1}{x^4+2x}+C\]

Answer 17

yup

Answer 18

one question which gets me confused. does the "du" get dropped when I do the anti derivative?

Answer 19

I always think i need to include it in the final simplified answer

Answer 20

what you should do to understand this trick of notation is take the derivative of your answer, and see why it gives you exactly what you want
it is the chain rule backwards

Answer 21

yes, the du gets dropped, it just says what the variable is, like the notation
\[\frac{d}{dx}[ax^2]\] for example

Answer 22

thank you