Question

Integral

Answer 1

\[\large \int\limits_{}^{}\frac{ x }{ (2x+1)^3 }\]

Answer 2

Oh, this is one of those ones where you'll turn into something like this,\[\large \int\limits\limits_{}^{}\frac{x}{(2x+1)^3}\quad=\quad \int\limits\frac{(u~stuff)}{(u)^3}~du\]And since the messy "stuff" is in the numerator,
you'll be able to easily break it down into multiple fractions.

Answer 3

We'll have `three pieces` to replace, instead of the two that you're used to.
\[\large\rm u=2x+1\]Solve for x.

Answer 4

So it would be\[\large x=\frac{ u-1 }{ 2 }\]

Answer 5

k good, and of course our last piece is the differential,\[\large\rm du=2dx\qquad\to\qquad \frac12du=dx\]

Answer 6

\[\large\rm \int\limits\frac{x}{(2x+1)^3}dx\quad=\quad \int\limits\frac{\frac12(u-1)}{u^3}~\frac12du\]

Answer 7

Do you see how that helps us?\[\large\rm =\frac14\int\limits\frac{u-1}{u^3}~du\]

Answer 8

Ohh, now I can split the fraction! I think I can do the rest now, that is quite clever, thanks.

Answer 9

Ya fun stuff c: