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

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The discussion focuses on solving the integral $$\int \frac{xe^{2x}}{(1+2x)^2}dx$$ using integration by parts, with participants sharing their methods and clarifying steps. One participant demonstrates the integration by parts technique, providing a detailed solution that simplifies to $$\frac{e^{2x}}{4+8x}+c$$. Other integrals are also discussed, including methods like partial fractions and substitution, with participants confirming the appropriateness of these techniques for different problems. The conversation emphasizes the importance of understanding the integration methods required for upcoming tests, particularly the necessity of using integration by parts as instructed by the instructor. Overall, the thread serves as a collaborative effort to clarify integration techniques and ensure comprehension ahead of assessments.
  • #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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