Solve Integration by Parts: Arctan(4t)

[MillerGenuine]

Homework Statement

$$\int arctan(4t)$$

Homework Equations

I know what the answer is to the problem but when i look at the solution i have no idea how they get from one step to the next.

The Attempt at a Solution

once we integrate by parts we get
$$1/4 U arctan(U) - 1/4 \int U/1+U^2$$ where U= 4t

from this they go to..

$$1/4 U arctan(U) - 1/8 \int 2U/1+U^2$$

clearly a 2 was multiplied inside and a 1/2 on the outside of the integral, but why? and how? because the next step shown is..

$$1/4 (4t arctan(4t) - 1/8 ln 16t^2 + c$$

which is the answer..i get that 4t was substituted back in for U, but i don't understand how and why the 2 was put in the integral and the 1/2 outside of it, and how in the last step the 2U somehow is gone?

 

[jgens]

once we integrate by parts we get
$$1/4 U arctan(U) - 1/4 \int U/1+U^2$$ where U= 4t

from this they go to..

$$1/4 U arctan(U) - 1/8 \int 2U/1+U^2$$

clearly a 2 was multiplied inside and a 1/2 on the outside of the integral, but why? and how? because the next step shown is..

You should use parantheses to make your work clearer. Anyway, note that (1+u²)' = 2u. The two is multiplied (and divided) so that you can apply the substitution rule for integration.