How to solve this?$$\int xe^{6x}dx $$

Question

anonymous · Answer

use integration by parts

anonymous · Answer

u=e^6x
du=6e^6x
dv=x
v=x^2/2

use integration by parts now

lgbasallote · Answer

i think it's easier to use $$u = x \implies du = dx$$
$$dv = e^{6x} \implies v = \frac{e^{6x}}6$$

parthkohli · Answer

I don't really know much about integration by parts. Please explain.

anonymous · Answer

parthkohli · Answer

Wait, I'm getting it...

anonymous · Answer

check this link..

parthkohli · Answer

$$\int udv = uv - \int vdu $$So that's why we take $u = x$ and $dv = e^{6x}dx$ right?

parthkohli · Answer

Hello, Ms. Chlorophyll!

parthkohli · Answer

What should I do next?

lgbasallote · Answer

you derive u and integrate dv

anonymous · Answer

Go with @lgbasallote's way, because what you want is to reduce the power of x

parthkohli · Answer

Ok...$$du = dx$$Because derivative of $x$ is 1.

lgbasallote · Answer

right.. and integral of dv?

parthkohli · Answer

And$$\int e^{6x}dx = {e^{6x} \over 6} $$

lgbasallote · Answer

yes.

lgbasallote · Answer

so now you have $$u = x$$
$$du = dx$$
$$v = \frac{e^{6}}6$$
you can now solve $$uv - \int vdu$$

lgbasallote · Answer

that v should be v = [e^(6x)]/6...you know what i mean

parthkohli · Answer

So,$$\int udv = uv - \int vdu $$...then$$\int xe^{6x}dx = {xe^{6x} \over 6} - \int {e^{6x} dx} $$??

parthkohli · Answer

I'm very slow at integration right now haha

anonymous · Answer

correct, now take integral of e^(6x) one more time!

anonymous · Answer

=xe^6x/6−e^6x/6

parthkohli · Answer

$$\int e^{6x} = {e^{6x} \over 6}$$

parthkohli · Answer

dx at the end...

parthkohli · Answer

$$\int xe^{6x}dx = {xe^{6x} \over 6} - {e^{6x} \over 6} $$$$\implies {x e^{6x} - e^{6x} \over 6} $$

parthkohli · Answer

$$\implies {e^{6x}(x - 1) \over 6} $$Is that correct?

parthkohli · Answer

and + c

parthkohli · Answer

Really? Is that really it!??!

parthkohli · Answer

Wow... can't believe I did it :|

anonymous · Answer

Remember when the terms in the integral have no relation of derivative for substitute, then think about Integral by parts!
Great job :)

parthkohli · Answer

Thank you, Ms. Chlorophyll!

anonymous · Answer

Also know, choose dv as the less complicated term!

parthkohli · Answer

OK, I get it... :)

anonymous · Answer

:) Practice makes perfect :)

parthkohli · Answer

OK, I am gonna practice!

parthkohli · Answer

Thank you!