Evaluating cosine function from ##-\infty## to ##\infty##

[happyparticle]

Homework Statement

Evaluate $\int_{-\infty}^{\infty} f(x)g(x) dx$
where $f(x) = \cos (ax) , g(x) = e^{-c^2x^2}$

Relevant Equations

$\int_{-\infty}^{\infty} \cos (ax) e^{-c^2x^2} dx$

Hi,
I have some question about evaluating a cosine function from $-\infty$ to $\infty$.
I saw for a cosine function evaluate from $-\infty$ to $\infty$ I can change the limits from 0 to $\infty$. I have a idea why, but I can't convince myself, furthermore, is it always the case no matter the cosine function?

Moreover, would it be appropriate to replace $cos(ax)$ for his complex equivalent, thus I will have only 2 exponentials function to deal with.

Thanks

 

[Orodruin]

First, you mean ”integrating”, not “evaluating”.

You can change limits to 0 to infinity (and multiply the result by 2!) because the integrand is even, not because a cosine is involved. For example $\cos(x + \pi/2)$ is not even.

Yes, you can use the identity with the complex exponential.

 

[happyparticle]

First, you mean ”integrating”, not “evaluating”.

That was I thought, however in my textbook the term evaluate the integral is used so I choose the word "evaluating". I was wondering if this is similar to integrate.

Thank you for the answer

 

[Orodruin]

That was I thought, however in my textbook the term evaluate the integral is used so I choose the word "evaluating". I was wondering if this is similar to integrate.

Thank you for the answer

You evaluate the integral to find its value. You do not evaluate the integrand. Saying ”evaluate an integral containing a cosine function” would be correct.