Integration by Parts: Solve $$\frac{xe^{2x}}{(1+2x)^2}$$

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SUMMARY

The forum discussion centers on solving the integral $$\int \frac{xe^{2x}}{(1+2x)^2}dx$$ using integration by parts. The solution involves setting $$u = \frac{-1}{1+2x}$$ and $$dv = xe^{2x}dx$$, leading to the final result of $$\frac{e^{2x}}{4+8x}+c$$. Additionally, participants discuss other integrals requiring techniques like partial fractions and reduction, emphasizing the importance of understanding integration methods for upcoming tests.

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  • #31
what do i do once i have $\frac{1}{64(sec\theta)^2}$
 
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  • #32
ineedhelpnow said:
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 $\theta$?
 

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