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

Click For Summary

Discussion Overview

The discussion revolves around the integration of the function $$\frac{xe^{2x}}{(1+2x)^2}$$ using integration by parts. Participants explore various integration techniques, including integration by parts, reduction, and substitution, while also discussing other integrals and methods relevant to their coursework.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants express uncertainty about how to apply integration by parts to the integral $$\int \frac{xe^{2x}}{(1+2x)^2}dx$$.
  • One participant proposes a method using integration by parts, providing a detailed calculation that leads to a specific result.
  • Another participant questions the validity of the methods used for different integrals, such as $$\int \frac{7x^2+x+24}{x^3+4x}dx$$ and $$\int \frac{e^{2x}}{6e^x+4}dx$$, suggesting alternative approaches like partial fractions and substitution.
  • There is a discussion about the necessity of using specific methods taught in class for test purposes, with participants expressing concern over which techniques are acceptable.
  • Some participants share their experiences with different integration techniques, such as reduction and substitution, while others seek clarification on how to set up integration by parts for their specific integrals.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to solve the integral $$\int \frac{xe^{2x}}{(1+2x)^2}dx$$, as multiple approaches are discussed, and some participants express confusion about the requirements for their test.

Contextual Notes

Participants mention the need to adhere to methods taught in class, indicating that some approaches may not be accepted in a testing context. There are also references to specific integrals and techniques that may not be fully resolved or clarified.

Who May Find This Useful

Students preparing for calculus tests, particularly those focused on integration techniques, may find this discussion relevant. Additionally, those interested in exploring various methods of integration, including integration by parts and substitution, could benefit from the shared insights.

  • #31
what do i do once i have $\frac{1}{64(sec\theta)^2}$
 
Physics news on Phys.org
  • #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 $\theta$?
 

Similar threads

High School Straightforward integration…
  • · Replies 27 ·
Replies
27
Views
2K
High School Integration by Parts, an introduction I get confused with
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Different Substitution for Solving an Integral
  • · Replies 6 ·
Replies
6
Views
3K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Gaussian integral by differentiating under the integral sign
  • · Replies 19 ·
Replies
19
Views
4K
Undergrad Did Math DF's integral calculator make a glaring mistake?
  • · Replies 8 ·
Replies
8
Views
3K
MHB What is the integral of $xe^{2x}$ divided by the square of $1+2x$?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Derive the orthonormality condition for Legendre polynomials
  • · Replies 30 ·
2
Replies
30
Views
2K
MHB Integrate by Parts: Solving (xe^(2x))/((1+2x)^2)
  • · Replies 5 ·
Replies
5
Views
2K
MHB What Is the Value of the Integral $\int_0^1 xe^x \, dx$?
  • · Replies 1 ·
Replies
1
Views
1K