How can I integrate x/(x^2 + ax + a^2)?

  • Thread starter Swatch
  • Start date
  • Tags
    Integrate
In summary, the conversation discusses the integration of x/(x^2 + ax + a^2) and various methods that have been attempted, including substitution and partial fractions. The suggestion to complete the square is given as a hint for solving the integral.
  • #1
Swatch
89
0
Hi. I want to integrate x/(x^2 + ax + a^2)
I tried substitution with u=x^2 then du =2x but that didn't work out neither did the substitution x^2 + ax
I thought of factorizing the denominator and using partial fractions, but I think that's not the way, can't figure out the factorization.
Could someone please give me a hint how to integrate this.
 
Last edited:
Physics news on Phys.org
  • #2
If there was a 2ax in the denominator instead of ax, it would've been easier :smile:
This is a typical example of an integral which will have an ln-part and an arctan-part. Do you see why/how?
 
  • #3
Swatch said:
Hi. I want to integrate x/(x^2 + ax + a^2)
I tried substitution with u=x^2 then du =2x but that didn't work out neither did the substitution x^2 + ax
I thought of factorizing the denominator and using partial fractions, but I think that's not the way, can't figure out the factorization.
Could someone please give me a hint how to integrate this.

TD's hint is best: complete the square!
 
  • #4
Compleated the square and everything works. Thanks guys.
 

1. What is the general formula for integrating x/(x^2 + ax + a^2)?

The general formula for integrating x/(x^2 + ax + a^2) is: ln|x^2 + ax + a^2|/2 + C, where C is the constant of integration.

2. How do you solve the integral of x/(x^2 + ax + a^2)?

To solve the integral of x/(x^2 + ax + a^2), you can use partial fractions or trigonometric substitution, depending on the values of a. Alternatively, you can use the general formula for integrating this type of function.

3. Can x/(x^2 + ax + a^2) be integrated using u-substitution?

Yes, you can use u-substitution to integrate x/(x^2 + ax + a^2) by setting u = x^2 + ax + a^2 and du = (2x + a)dx.

4. Is there a specific value of a that makes the integration of x/(x^2 + ax + a^2) easier?

Yes, if a = 0, the integral of x/(x^2 + ax + a^2) simplifies to ln|x| + C, which is the natural logarithm of the absolute value of x plus a constant.

5. What are some real-world applications of integrating x/(x^2 + ax + a^2)?

The function x/(x^2 + ax + a^2) can be used to model the velocity of an object undergoing simple harmonic motion. It is also used in calculating the electric potential of a ring of charge, as well as in solving certain differential equations in physics and engineering.

Similar threads

Find the value of the definite integral
    • Calculus and Beyond Homework Help
Replies
8
Views
763
Differentiate the given integral
    • Calculus and Beyond Homework Help
Replies
15
Views
786
Fourier Transform - Solutions Error?
    • Calculus and Beyond Homework Help
Replies
5
Views
357
Integral using substitution x = -u
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
How To Integrate 1/[sqrt (x^2 + 3x + 2)] dx?
    • Calculus and Beyond Homework Help
2
Replies
44
Views
4K
Solve the given problem that involves integration
    • Calculus and Beyond Homework Help
Replies
4
Views
739
Wolfram gave me one answer, examiners gave me another
    • Calculus and Beyond Homework Help
Replies
4
Views
622
Integrals that keep me up at night
    • Calculus and Beyond Homework Help
Replies
22
Views
1K
A doubt in Partial fraction decomposition
    • Calculus and Beyond Homework Help
Replies
8
Views
948
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
    • Calculus and Beyond Homework Help
Replies
7
Views
706

Hot Threads

Recent Insights

Back
Top