Question

Requesting help with U-substitution with integrals.

Answer 1

It's the indefinite integral of:
d x\[(3+\ln x)^2(2-\ln x) \over 4x\]

Answer 2

\[\int\frac{(3+\ln x)^2(2-\ln x)}{4x}~dx\]
Let \(u=\ln x\), so that \(du=\dfrac{1}{x}~dx\):
\[\int\frac{(3+u)^2(2-u)}{4}~du\]

Answer 3

Then I can factor out a 1/4 and I get\[\frac{ 1 }{ 4 } \int\limits (3+u)^2(2-u)\] right?

Answer 4

Yes of course.

Answer 5

After this I don't know what to do. Do I multiply the two together or integrate by parts?

Answer 6

Integrating by parts may take years lol. Please expand everything under that integral.

Answer 7

Or, if you really want to punish yourself, you can make another substitution. But yeah, expanding is the way to go here.

Answer 8

so then i get \[\frac{ 1 }{ 4 } \int\limits 18 +3u-4u^2-u^3\]

Answer 9

and the antiderivative is \[\frac{ 1 }{ 4 } 18u+\frac{ 3 }{ 2 }u^2-\frac{ 4 }{ 3 }u^3-\frac{ 1 }{ 4 }u^4\]

Answer 10

+c

Answer 11

and then I plug in u?

Answer 12

Yes. Write everything in a bracket with 1/4 outside so its clearer.

Answer 13

Alright So that's it then. Thank you so much.

Answer 14

Welcome