# Thread: Laplace transform to calculate an integral.

1. ## Laplace transform to calculate an integral.

Use Laplace transform (with respect to t)to calculate the integral

I=\int([\cos(tx)/(x^2+a^2)]dx t\geqslant 0

[IMG]file:///C:/Users/Jamie/AppData/Local/Temp/msohtmlclip1/01/clip_image002.jpg[/IMG]

2. Originally Posted by sublim25
Use Laplace transform (with respect to t)to calculate the integral

I=\int([\cos(tx)/(x^2+a^2)]dx t\geqslant 0

[IMG]file:///C:/Users/Jamie/AppData/Local/Temp/msohtmlclip1/01/clip_image002.jpg[/IMG]

So you have

$\displaystyle I =\int\frac{\cos(tx)}{x^2+a^2}dx$

but what are the limits of integration?

3. Limits are from 0 to infinity.

4. Let

$\displaystyle I(t) =\int_{0}^{\infty}\frac{\cos(tx)}{x^2+a^2}dx$

Then

$\displaystyle \mathcal{L}\{I\} =\int_{0}^{\infty}\frac{\mathcal{L}\{\cos(tx)\}}{x ^2+a^2}dx=\int_{0}^{\infty}\frac{s}{s^2+x^2}\cdot \frac{1}{x^2+a^2}dx$

Now by partial factions we get

$\displaystyle \mathcal{L}\{I\} =\frac{s}{s^2-a^2}\int_{0}^{\infty}\left( \frac{1}{x^2+a^2}-\frac{1}{x^2+s^2}\right)dx=\frac{\pi}{2}\left( \frac{s}{a(s^2-a^2)}-\frac{1}{s^2-a^2}\right)$

Now if you take the inverse Laplace transform we get

$\displaystyle I=\frac{\pi}{2a}\cosh(at)-\frac{\pi}{2}\sinh(at)$

5. Can you please explain how you get the partial fractions to come out to this?

6. Originally Posted by sublim25
Can you please explain how you get the partial fractions to come out to this?
$\displaystyle \frac{s}{(x^2+a^2)(x^2+s^2)}=\frac{Ax+B}{x^2+a^2}+ \frac{Cx+D}{x^2+s^2}$

$\displaystyle s =(Ax+B)(x^2+s^2)+(Cx+D)(x^2+a^2)$

Now expand all of this out to get

$\displaystyle s=Ax^3+Bx^2+s^2Ax+s^2B + Cx^3+Dx^2+a^2Cx+a^2D$

$\displaystyle s=(A+C)x^3+(B+D)x^2+(s^2A+a^2C)x+(s^2B+a^2D)$

So this gives 4 equations in the 4 unknowns A,B,C,D

$\displaystyle A+C=0 \quad B+D=0 \quad s^2A+a^2C=0 \quad s^2B+a^2D=s$

dont forget that s and a are constants. Now just solve this system.

7. I am not sure what I am doing wrong, but when I solve the system, everything is cancelling out.

8. Originally Posted by sublim25
I am not sure what I am doing wrong, but when I solve the system, everything is cancelling out.
I don't know either. Please post what you have done. Note that both A and C are equal to 0.