Thread: integral cosine and exponential

1. integral cosine and exponential

I'm trying to solve the following integral

int_0^{+infintity} 1/(1+a^2*x^2)*exp(-b^2*x^2)*cos(c*x) dx

where a,b and c are real.

thanks a lot

Marco

2. Originally Posted by mathmarco
I'm trying to solve the following integral

int_0^{+infintity} 1/(1+a^2*x^2)*exp(-b^2*x^2)*cos(c*x) dx

where a,b and c are real.

thanks a lot

Marco
$\int_0^{\infty}\frac{e^{-b^2x^2}}{1+a^2x^2}\cos(cx)~dx$?

3. yes, it is exactly that.

I will use \math next time

4. Originally Posted by mathmarco
I'm trying to solve the following integral

int_0^{+infintity} 1/(1+a^2*x^2)*exp(-b^2*x^2)*cos(c*x) dx

where a,b and c are real.

thanks a lot

Marco
Well first off I am pretty sure that we must have that $b\in\mathbb{N}$

Otherwise the integral would diverge.

5. I though it was not necessary to have $b\in N$
It is not sufficient that $b\in R$

why do you think it will diverge? Does not the following integral converge?
$\int_0^{+\infty} e^{-b^2 x^2} dx$

I'm maybe missing something...

6. Originally Posted by mathmarco
I though it was not necessary to have $b\in N$
It is not sufficient that $b\in R$

why do you think it will diverge? Does not the following integral converge?
$\int_0^{+\infty} e^{-b^2 x^2} dx$

I'm maybe missing something...
$\int_0^{\infty}e^{-b^2x^2}~dx=\int_0^{\infty}e^{-(bx)^2}$

Now let $u=bx\Rightarrow\frac{u}{b}=x$

So $dx=\frac{1}{b}$

so we have that when $x\to\infty\Rightarrow{u\to\infty}$

and when

$x\to{0}\Rightarrow{u\to{0}}$

So we have

$\frac{1}{b}\int_0^{\infty}e^{-u^2}~du=\frac{\sqrt{\pi}}{2b}$

Yeah you are right, I misread it at first as $b$ not $b^2$

7. ok... good. I'm still trying to solve it but I always end up to nothing good!!

8. Originally Posted by mathmarco
ok... good. I'm still trying to solve it but I always end up to nothing good!!
I can do little more than show it converges, are you sure it is integralbe? Have you tried differentiation under the integral sign after a sub of $bx=u$?

$\forall{x}\in\mathbb{R}\quad{-1\leq\cos(cx)\leq{1}}$

So

$-\int_0^{\infty}\frac{e^{-b^2x^2}}{1+a^2x^2}~dx\leq\int_0^{\infty}\frac{\cos (cx)\cdot{e^{-b^2x^2}}}{1+a^2x^2}~dx\leq\int_0^{\infty}\frac{e^{-b^2x^2}}{1+a^2x^2}~dx$

Or in other words

$\bigg|\int_0^{\infty}\frac{\cos(cx)e^{-b^2x^2}}{1+a^2x^2}~dx\bigg|\leq\int_0^{\infty}\fra c{e^{-b^2x^2}}{1+a^2x^2}~dx$

and obviously since

$0\leq\int_0^{\infty}\frac{e^{-b^2x^2}}{1+a^2x^2}~dx\ll\int_0^{\infty}\frac{dx}{1 +a^2x^2}$

Now

$\int_0^{\infty}\frac{dx}{1+(ax)^2}$

Let $ax=\varphi\Rightarrow\frac{\varphi}{a}=x$

So $dx=\frac{1}{a}$

And $\text{As }x\to\infty\Rightarrow\varphi\to\infty$

and $\text{As }x\to{0}\Rightarrow\varphi\to{0}$

So we have

$\frac{1}{a}\int_0^{\infty}\frac{d\varphi}{1+\varph i^2}=\frac{1}{a}\arctan\left(\varphi\right)\bigg|_ 0^{\infty}=\frac{\pi}{2a}$

Therefore $\int_0^{\infty}\frac{e^{-b^2x^2}}{1+a^2x^2}~dx$

So we can finally conclude that

$\int_0^{\infty}\frac{e^{-b^2x^2}\cos(cx)}{1+a^2x^2}~dx\quad\text{converges} ~\forall~(a,b,c)\in\mathbb{R}$

Furthermore we see that

$\bigg|\int_0^{\infty}\frac{e^{-b^2x^2}\cos(cx)}{1+a^2x^2}~dx\bigg|\leq\int_0^{\in fty}\frac{e^{-b^2x^2}}{1+a^2x^2}~dx\leq\frac{\pi}{2a}$

Hope this has been some help.