Integration: 1/((x^(1/2)-x^(1/3))

  • Context: Undergrad 
  • Thread starter Thread starter Bohn507
  • Start date Start date
  • Tags Tags
    Integration
Click For Summary

Discussion Overview

The discussion revolves around the integration of the function \( \frac{1}{\sqrt{x} - \sqrt[3]{x}} \). Participants explore various approaches to solve the integral, including substitutions and transformations. The scope includes mathematical reasoning and integration techniques.

Discussion Character

  • Mathematical reasoning, Homework-related

Main Points Raised

  • One participant expresses uncertainty about how to begin solving the integral.
  • Another participant clarifies the problem statement and provides a link to an external tool for assistance.
  • A third participant thanks the previous contributor for the clarification.
  • A later reply proposes a substitution \( x = u^6 \) and outlines the steps to transform the integral, leading to a more manageable form for integration.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach, as multiple methods are presented, and some express uncertainty about the initial steps.

Contextual Notes

Some assumptions about the validity of the substitution and the steps involved in integration remain unverified, and there may be dependencies on the definitions of the functions involved.

Bohn507
Messages
4
Reaction score
0
I have no idea where to start with this. Sorry about the format, I don't know where to make it into an easier to read style.

1/((x^(1/2)-x^(1/3))
 
Physics news on Phys.org
The problem is: [itex]\int \dfrac{1}{\sqrt{x} - \sqrt[3]{x}} dx[/itex]

Here, http://www.wolframalpha.com/input/?i=integrate+1%2F((x^(1%2F2)-x^(1%2F3))
Click on show steps and that's it.

See LaTeX for formatting your equations here.
 
Thank you!
 
Or, you could do as follows:
Introduce:
[tex]x=u^{6}\to\frac{dx}{du}=6u^{5}\to{dx}=6u^{5}du[/tex]
Then,
[tex]\int\frac{dx}{\sqrt{x}-\sqrt[3]{x}}=\int\frac{6u^{5}}{u^{3}-u^{2}}du=\int\frac{6u^{3}}{u-1}du=6\int({u}^{2}+u+1+\frac{1}{u-1})du[/tex]
which is easily integrated.
 

Similar threads

Undergrad Negative area above x-axis from integrating x^2?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad ##f(2x)=f^2(x)-2f(x)-1/2## then find ##f(3)##
  • · Replies 17 ·
Replies
17
Views
4K
Undergrad Integration with different infinitesimal intervals
  • · Replies 31 ·
2
Replies
31
Views
5K
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 How Do You Integrate 1/√(x^3 + x^2 + x + 1) dx?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Another Integration Formula
  • · Replies 6 ·
Replies
6
Views
2K
High School Why is this definite integral a single number?
  • · Replies 20 ·
Replies
20
Views
4K
High School Straightforward integration…
  • · Replies 27 ·
Replies
27
Views
3K
Undergrad Derive the orthonormality condition for Legendre polynomials
  • · Replies 30 ·
2
Replies
30
Views
2K