# Homework Help: Laplace Transform Integral

1. Dec 31, 2013

### pierce15

1. The problem statement, all variables and given/known data

$$\int_0^\infty \frac{\sin xt}{x} \, dt$$

2. Relevant equations

3. The attempt at a solution

$$= \int_0^\infty L(\sin xt) \, dp$$

$$= \int_0^\infty \frac{x}{p^2 + x^2} \, dp$$

$$= x \int_0^\infty \frac{dx}{p^2 + x^2} \, dp$$

p = x tan theta:

$$= x \int_0^{\pi/2} \frac{ \sec^2 \theta}{x^2 \sec^2 \theta} \, d\theta$$

$$= \frac{1}{x} \cdot \frac{\pi}{2}$$

My textbook says that the answer should be exactly pi /2. What did I do wrong?

Last edited: Dec 31, 2013
2. Dec 31, 2013

### pierce15

Never mind, I found the problem: I forgot to include the x in dx = x sec^2 theta d theta. However, while we're here, I have another textbook problem:

$$\int_0^ \infty \frac{ \cos xt}{1 + t^2} \, dt$$

I have noticed that this is expressible as

$$\int_0^\infty \cos xt \cdot L[ \sin x ] \, dt$$

Is that the right first step? I'm not sure where to go from here