Integration by parts for xe^(2x)/(1+2x)^2

[ineedhelpnow]

what do i do once i have $\frac{1}{64(sec\theta)^2}$

 

[MarkFL]

what do i do once i have $\frac{1}{64(sec\theta)^2}$

Well, you need to change your differential and limits in accordance with the substitution...we have let:

$$\frac{1}{y}=8\tan(\theta)\,\therefore\,y=\frac{1}{8}\cot(\theta)\,\therefore\,dy=-\frac{1}{8}\csc^2(\theta)\,d\theta$$

Now, from our substitution, we find:

$$\theta=\tan^{-1}\left(\frac{1}{8y}\right)$$

and so we use this to change our limits from $y$'s to $\theta$'s.

Can you put all of this together to express the remaining integral in terms of $\theta$?