Integration By Parts: Solving int.arctan(2x)dx for Calculus Homework

[mike01]

Integration By Parts?

Homework Statement

int.arctan(2x)dx

Homework Equations

Integration By Parts

The Attempt at a Solution

In the attached image is the original problem with the ansewer I came up with using integration by parts and then a v=sub. later in the problem I did not want to post additional steps because it turned out to be a longer problem than I thought Just curious if someone could confirm my ansewer and if it is incorrect I will post the work to help see where I messed up. thanks a ton.

 

[Dick]

I differentiated your antiderivative and I didn't get arctan(2x). If you want to check your work in the future, you could try that too. You can often get a clue where you messed up by looking at that as well.

 

[mike01]

thanks I will see if I can figure it out.

 

[crims0ned]

Yeah i almost got the same thing, except for the (1/4) looks like just a u-sub

Integral of arctan(2x) dx... u=2x du=2dx dx=(1/2)du

so now we have (1/2) integ arctan(u) du

leave the (1/2) out in front as a constant and I got u*arctan(u)-ln(sqrt(1+u^2))

plug everything back in and i got x*arctan(2x)-(1/2)ln(sqrt(1+4x^2)) ... but I just used a table for arctan(u)