Fraction of integrals with different variables

In summary, for evaluating the given expression without using trigonometric substitution, you can use a standard integral and a normal substitution, and then solve them separately. For the second expression, you can use the integral \int \frac{1}{1 + x^2} \, dx = \tanh(x) and a non-trig substitution, and then solve them separately. It is important to be careful with the notation and use the correct integral function.
  • #1
huey910
36
0
how would one evaluate this without using trig substitution? Is it possible to make one integral out of this?

{int[(y^2 + a1^2)^-1]dy +c1}/{int[(x^2 + a2^2)^-1]dx +c2} +c3

the numbers behind the 'a's and 'c's are supposed to be subscripts.

Also, how would one deal with this:

{int[(y^2 + a1^2)^-1]dy +c1}/(a tan{int[(x^2 + a2^2)^-1]dx +c2}) +c3


Please advise
 
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  • #2
You can use the standard integral
[tex]\int \frac{1}{1 + x^2} \, dx = \tanh(x)[/tex]
and a "normal" (non-trig) substitution, and then you can do them separately.
 
  • #3
CompuChip said:
You can use the standard integral
[tex]\int \frac{1}{1 + x^2} \, dx = \tanh(x)[/tex]
and a "normal" (non-trig) substitution, and then you can do them separately.

That doesn't seem right. Isn't that integral tan(x)+C, and not tanh(x)+C?
 
  • #4
Char. Limit said:
That doesn't seem right. Isn't that integral tan(x)+C, and not tanh(x)+C?

Or rather atan(x)+C...
 
  • #5
micromass said:
Or rather atan(x)+C...

Touche.
 
  • #6
Heh, double fail :-)
I knew it was something with tan and an extra letter! Thanks micromass :)
 

What is a fraction of integrals with different variables?

A fraction of integrals with different variables refers to the ratio of the number of integrals involving different variables to the total number of integrals in a given mathematical expression.

Why is it important to consider the fraction of integrals with different variables?

It is important to consider the fraction of integrals with different variables because it can affect the complexity and difficulty of solving a mathematical expression. Higher fractions of integrals with different variables may require more advanced techniques and longer calculations.

How do you calculate the fraction of integrals with different variables?

To calculate the fraction of integrals with different variables, first count the total number of integrals in the expression. Then, count the number of different variables present in each integral. Divide the number of integrals with different variables by the total number of integrals to get the fraction.

What does a lower fraction of integrals with different variables indicate?

A lower fraction of integrals with different variables indicates that the mathematical expression is simpler and may be easier to solve. It may also indicate that the expression has a higher level of symmetry or that the variables are related in some way.

Can the fraction of integrals with different variables be reduced or manipulated?

Yes, the fraction of integrals with different variables can be reduced or manipulated by using mathematical techniques such as variable substitution or integration by parts. However, the resulting expression may not always be equivalent to the original one.

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