Question

How would I integrate (1-2x)^3?

Answer 1

do you know how to integrate something like:
\[-\frac{1}{2}u^3\] w.r.t to u ?

Answer 2

\[u=1-2x \\ \frac{du}{dx}=-2 \\ du=-2 dx \\ \frac{-1}{2} du=dx \\ \int\limits (1-2x)^3 dx=\int\limits u^3 \frac{-1}{2} du=\frac{-1}{2} \int\limits u^3 du\]

Answer 3

I understand until the 4th step. Would you be able to explain it to me?

Answer 4

the 4th line I wrote?
I divide both sides by -2
or the equivalent multiplied both sides by -1/2

Answer 5

Okay. :) Then do we use the chain rule for the 5th line you wrote?

Answer 6

yeah

Answer 7

Do you need to simplify -1/2 (1-2x)^3 to get the final answer?

Answer 8

well you still need to integrate u^3

Answer 9

w.r.t. u

Answer 10

So the next step would be -1/2 x u^4/4? And then I substitute 1-2x for u?

Answer 11

yep

Answer 12

and that one x is a multiplying symbol right?

Answer 13

Yes

Answer 14

\[\frac{-1}{2} \cdot \frac{u^4}{4}+C\]
and yes you just replace u with (1-2x)

Answer 15

and you could rewrite as
\[\frac{-(1-2x)^4}{8}+C\]
I would do nothing else
expanding that makes it look nastier

Answer 16

So the final answer is -((1-2x)^4))/8 + c?

Answer 17

Thank you! :)

Answer 18

np