How to Approach the Integration of Complex Functions with Limits from -3 to 3?

[lap]

Integrate ( (sqrt (x^2 - 9))/x )( exp x^2 )( cos 7x )( sin(x^4 + 5x^2 + 100) ) dx
with upper limit = 3 and lower limit = -3

I have tried to use integration by part and set u = ( (sqrt (x^2 - 9))/x )( exp x^2 ) and
dv = ( cos 7x )( sin(x^4 + 5x^2 + 100) ) dx

 

[adjacent]

What have you tried so far?

 

[Ray Vickson]

Integrate ( (sqrt (x^2 - 9))/x )( exp x^2 )( cos 7x )( sin(x^4 + 5x^2 + 100) ) dx
with upper limit = 3 and lower limit = -3

I have tried to use integration by part and set u = ( (sqrt (x^2 - 9))/x )( exp x^2 ) and
dv = ( cos 7x )( sin(x^4 + 5x^2 + 100) ) dx

But I don't know how to integrate the dv

I very much doubt there is any closed-form formula for the antiderivative, so you probably need to contemplate numerical integration for the general case of $\int_a^b f(x) \, dx$. However, before doing that, sit down and think carefully about your specific problem.

 

[lap]

How to integrate ( (sqrt (x^2 - 9))/x )( exp x^2 )( cos 7x )( sin(x^4 + 5x^2 + 100) ) dx ?

 

[Ray Vickson]

How to integrate ( (sqrt (x^2 - 9))/x )( exp x^2 )( cos 7x )( sin(x^4 + 5x^2 + 100) ) dx ?

I have already told you it cannot be done with formulas---even very long ones having billions of complicated terms and taking millions of pages to write out. However, that was not your original question: you wanted $\int_{-3}^3 f(x) \, dx$. As I suggested, think hard about the problem first.

 

[lap]

The answer is 0 ?

 

[dirk_mec1]

The answer is correct but can you prove it?

 

[lap]

I know the answer is 0 because the positive area canceled the negative area but I don't know how to prove it

 

[Dick]

I know the answer is 0 because the positive area canceled the negative area but I don't know how to prove it

If f(x) is that big expression you are integrating, can you prove that f(-x)=(-f(x))? Then show $\int_{-a}^0 f(x) dx = -\int_{0}^a f(x) dx$.

 

[lap]

I proved that f(-x)=(-f(x)) and solved it. Thank you very much !

 

[HallsofIvy]

In order to use symmetry here you must also show that this is not an improper integral. Your integrand is a fraction with sin(x^4+ 5x^2+ 100) in the denominator. Can you show that this never 0 for x between -3 and 3?

 

[Saitama]

In order to use symmetry here you must also show that this is not an improper integral. Your integrand is a fraction with sin(x^4+ 5x^2+ 100) in the denominator. Can you show that this never 0 for x between -3 and 3?

There is an x in the denominator instead of sine. Moreover, the function doesn't seem to defined within the given limits.