How to integrate 1/(x^2 +1)^2, does partial fractions work?

In summary, integrating 1/(x2 + 1)2 can be solved using trigonometric substitution by setting x=tan(u) and using the general trig substitution laws. Another method is using partial fractions in terms of complex linear fractions, which can then be rewritten in a real form using the relation between logarithm and arctan.
  • #1
timjones007
10
0
how do you integrate 1/(x2 + 1)2 ?


i have tried integration by partial fractions but when you set 1 equal to (Ax +B)(x2+1) + (Cx + D) this leads to A=B=C=0 and D=1 which just gives you the original equation
 
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  • #2


Try subbing x=tan(u), dx=1/cos^2(u)
 
  • #3
  • #4


Partial fraction in terms of the complex linear fractions 1/(x±i) and 1/(x±i)^2 will also work. You can then take together terms with their complex conjugate and then rewrite everything in a manifestly real form (you need to use the relation between the logarithm and arctan etc.).
 
  • #5


ah thank you that works
 

Related to How to integrate 1/(x^2 +1)^2, does partial fractions work?

1. What is the general method for integrating 1/(x^2 + 1)^2 using partial fractions?

The general method for integrating 1/(x^2 + 1)^2 using partial fractions involves first factoring the denominator into linear factors, then writing the partial fraction decomposition, and finally solving for the unknown coefficients.

2. Can any rational function be integrated using partial fractions?

Yes, any rational function can be integrated using partial fractions as long as the degree of the numerator is less than the degree of the denominator.

3. What is the benefit of using partial fractions to integrate 1/(x^2 + 1)^2?

The benefit of using partial fractions to integrate 1/(x^2 + 1)^2 is that it can simplify the integration process and make it easier to solve. It also allows for the use of known integration formulas for simpler terms.

4. Are there any special cases or restrictions when using partial fractions to integrate 1/(x^2 + 1)^2?

Yes, when the partial fraction decomposition results in repeated factors in the denominator, special cases or additional steps may be required. Additionally, if the denominator has complex or imaginary roots, the partial fraction decomposition will involve complex coefficients.

5. Is there an alternative method for integrating 1/(x^2 + 1)^2 besides partial fractions?

Yes, there are alternative methods for integrating 1/(x^2 + 1)^2 such as using trigonometric substitutions or completing the square. However, partial fractions is typically the most efficient and straightforward method for integration in this case.

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