# Thread: Integral Calculation - Urgent

1. ## Integral Calculation - Urgent

Hello,

I need to show that an integral from minus infinity to infinity is smaller than infinity. All the expressions that are integrated are positive. What should I do? Can you recommend me some theorems that I can use? Thank you in advance!

2. Originally Posted by joreto_co
Hello,

I need to show that an integral from minus infinity to infinity is smaller than infinity. All the expressions that are integrated are positive. What should I do? Can you recommend me some theorems that I can use? Thank you in advance!
My best recommendation (for the moment at least) is that you post the integral here.

3. Please try to post the image again, it isn't uploaded.

4. Here in the attachment is what I have to prove

5. $\int_{-\infty}^{\infty} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx$

As |x| and x^2 are positive, we can write this as,

$2\int_{0}^{\infty} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx$

So checking the convergence of $\int_{0}^{\infty} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx$ is enough.

$\int_{0}^{\infty} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx = \left [ \int_{0}^{1} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx \right ] ~+~ \left [ \int_{1}^{\infty} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx \right ]$

We can easily prove that $\int_{1}^{\infty} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx$ converges.

For $1\leq x \leq \infty$,
$0\leq \frac{\ln x \cdot a \cdot e^{-\frac{a}{x^2}}}{x^3} \leq \frac{\ln x \cdot a}{x^3}$

$\int_1^{\infty} \frac{\ln x \cdot a}{x^3}~dx = \frac{a}{4}$

So $\int_{1}^{\infty} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx$ converges too.

Now prove the convergence of $\int_{0}^{1} \frac{|\ln |x|| \cdot a \cdot e^{-\frac{a}{x^2}}}{|x|^3}~dx$ and it'll be over.
(It looks a little long and I can't think of anything easy at the moment..)

6. And does someone have an ideal for the second equation?