# Thread: Find integral for this

1. ## Find integral for this

Hi i'm lookin for the integral of t*exp(-abs(t))sin(wt) dt with limits at +infinity and -infinity

Anyone available to help?

2. Well, I have a solution but no steps..

$\displaystyle \frac{4w}{(1+w^2)^2}$
Greets,Leslon

3. Hi
Originally Posted by bcvw85
Hi i'm lookin for the integral of t*exp(-abs(t))sin(wt) dt with limits at +infinity and -infinity
First, let's get rid off the absolute value. As $\displaystyle t\mapsto t\sin(\omega t)\exp(-|t|)$ is an even function,

$\displaystyle I=\int_{-\infty}^{\infty}t\sin(\omega t)\exp(-|t|)\,\mathrm{d}t=2\int_0^{\infty}t\sin(\omega t)\exp(-t)\,\mathrm{d}t$

Then, one can use the expression of the sine in terms of exponential functions : $\displaystyle \sin(\omega t)=\frac{\exp(i\omega t)-\exp(-i\omega t)}{2i}$ which gives

\displaystyle \begin{aligned} I&=\int_0^{\infty}t\left(\frac{\exp(i\omega t)-\exp(-i\omega t)}{2i}\right)\exp(-t)\,\mathrm{d}t\\ &=\int_0^{\infty}t\left(\frac{\exp((i\omega-1) t)-\exp(-(1+i\omega) t)}{2i}\right)\,\mathrm{d}t\end{aligned}

Using integration by parts one can get rid off the $\displaystyle t$ which is outside the exponential functions.

4. Originally Posted by flyingsquirrel
Hi

First, let's get rid off the absolute value. As $\displaystyle t\mapsto t\sin(\omega t)\exp(-|t|)$ is an even function,

$\displaystyle I=\int_{-\infty}^{\infty}t\sin(\omega t)\exp(-|t|)\,\mathrm{d}t=2\int_0^{\infty}t\sin(\omega t)\exp(-t)\,\mathrm{d}t$

Then, one can use the expression of the sine in terms of exponential functions : $\displaystyle \sin(\omega t)=\frac{\exp(i\omega t)-\exp(-i\omega t)}{2i}$ which gives

\displaystyle \begin{aligned} I&=\int_0^{\infty}t\left(\frac{\exp(i\omega t)-\exp(-i\omega t)}{2i}\right)\exp(-t)\,\mathrm{d}t\\ &=\int_0^{\infty}t\left(\frac{\exp((i\omega-1) t)-\exp(-(1+i\omega) t)}{2i}\right)\,\mathrm{d}t\end{aligned}

Using integration by parts one can get rid off the $\displaystyle t$ which is outside the exponential functions.
You are making hard work of this, see the other thread

RonL

5. Originally Posted by bcvw85
Hi i'm lookin for the integral of t*exp(-abs(t))sin(wt) dt with limits at +infinity and -infinity

Anyone available to help?
$\displaystyle \int_{-\infty}^{\infty}te^{-|t|}\sin(wt)\,dt$

Write it as two separate integrals:

$\displaystyle \int_{-\infty}^{0}te^{t}\sin(wt)\,dt+\int_{0}^{\infty}te^ {-t}\sin(wt)\,dt$

Apply integration by parts:

let $\displaystyle u=t$ and $\displaystyle dv=e^{t}\sin(wt)\,dt$

Therefore, $\displaystyle \,du=\,dt$ and $\displaystyle v=\frac{1}{w^2+1}\cdot\left(e^t\sin(wt)-e^t\cos(wt)\right)$ (you get v by using integration by parts...again...

Plugging into the integration by parts formula:

$\displaystyle \left.\left[\frac{te^t}{w^2+1}\cdot\left(\sin(wt)-\cos(wt)\right)\right]\right|_{-\infty}^{0}-\frac{1}{w^2+1}\int_{-\infty}^{0}\left(e^t\sin(wt)-e^t\cos(wt)\right)\,dt$

Before we move on, note that $\displaystyle \left.\left[\frac{te^t}{w^2+1}\cdot\left(\sin(wt)-\cos(wt)\right)\right]\right|_{-\infty}^{0}$ converges to 0.

Thus, we have left:

$\displaystyle -\frac{1}{w^2+1}\int_{-\infty}^{0}\left(e^t\sin(wt)-e^t\cos(wt)\right)\,dt$

Apply integration by parts (again) for these two integals:

$\displaystyle -\frac{1}{w^2+1}\int_{-\infty}^{0}e^t\sin(wt)+\frac{1}{w^2+1}\int_{-\infty}^{0}e^t\cos(wt)\,dt$

I will leave steps out, hoping that you can do it on your own.

Now, the integrals should become $\displaystyle \left.\left[\frac{2we^t\cos(wt)}{(w^2+1)^2}\right]\right|_{-\infty}^{0}-\left.\left[\frac{(w^2-1)e^t\sin(wt)}{(w^2+1)^2}\right]\right|_{-\infty}^{0}$

$\displaystyle \left[\frac{2w}{(w^2+1)^2}-0\right]-\left[0-0\right]=\color{red}\boxed{\frac{2w}{(w^2+1)^2}}$

Since we only evaluated half of the integral, double it to get the answer:

$\displaystyle \therefore \int_{-\infty}^{\infty}te^{-|t|}\sin(wt)\,dt=\color{red}\boxed{\frac{4w}{(w^2+ 1)^2}}$