# Please help me to solve this integration problem

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

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## Answers and Replies

adjacent
Gold Member
What have you tried so far?

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
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.

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

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
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.

The answer is 0 ?

The answer is correct but can you prove it?

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

Dick
Science Advisor
Homework Helper
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##.

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

HallsofIvy
Science Advisor
Homework Helper
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?

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.