How to integrate function 1/(x^2+a) ?

In summary, the conversation is about calculating the integral of 1/(x^2+a) dx, where a is a real number greater than 0. The speaker is looking for a primitive function and has found that for a=1, the primitive function is arctan(x) + C. They are unsure of how it would change for different values of a and are seeking help. The summary also includes the speaker's suggestion to use a substitution to solve the integral.
  • #1
Holali
9
0
Hi,
I would like to calculate one integral. I just want to get primitive function, not definite integral.

∫ 1/(x^2+a) dx

where a is real number and >0

I only found that

∫ 1/(x^2+1) dx ,
its arctan(x) + C,
but i don't know how it is with different 'a' values.
Thanks for help!
 
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  • #2
First, this should not have been posted under "differential equations". I will move it to "Calculus and Analysis".

Second, let [itex]y= x/\sqrt{a}[/itex] so that [itex]x= \sqrt{a}y[/itex] and [itex]dx= \sqrt{a}dy[/itex]. The integral becomes
[tex]\int\frac{1}{ay^2+ a}(\sqrt{a}dy)= \frac{\sqrt{a}}{a}\int \frac{1}{y^2+1}dy[/tex].
 

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

The process for integrating 1/(x^2+a) involves using the substitution method, where u = x^2 + a. This will result in the integral becoming 1/u, which can be integrated using the natural logarithm function.

2. Can the integral of 1/(x^2+a) be solved without using substitution?

No, the integral of 1/(x^2+a) cannot be solved without using substitution. This is because the integrand involves a quadratic term and substitution is necessary to simplify the integral.

3. What are the limits of integration for integrating 1/(x^2+a)?

The limits of integration for integrating 1/(x^2+a) depend on the specific problem or application. They can be determined by considering the bounds of the function or by using the fundamental theorem of calculus.

4. Are there any special cases or exceptions when integrating 1/(x^2+a)?

Yes, if the value of a is equal to 0, the integral becomes 1/x^2 which can be solved using the power rule. However, for all other values of a, substitution will still be necessary.

5. What are some real-world applications of integrating 1/(x^2+a)?

Integrating 1/(x^2+a) has many applications in physics, engineering, and economics. It can be used to calculate the area under a curve, find the center of mass of an object, or determine the value of a series in finance.

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