This question came out for my exam.
Impulse response h(t) = u(t) - u(t-1)
Find output when input is x(t) = (t^2 + t)u(t)

Do I use the equation ∫x(τ)h(t-τ)dτ from 0 to infinity or do I use laplace transform?

Since u(t) - u(t-1) is one from 0 to 1, I reduced the integral to the limits 0 to 1 and continued on from there. Am I right or wrong?

However, when I use laplace transform I get a totally diff answer. Which is right?

 

The convolution is not properly carried out in the time domain. more specifically, limits of the integral are wrong.
convolution is made by shifting the flipped version of one signal by t, then integrate the product of this signal with the other signal for all values of t.
as u've chosen to shift h, the output is divided into two intervals:
Partial overlapping: 0<=t<1, limits are (0) to (t).
complete overlapping : 1<= t, limits are (t-1) to (t)
otherwise, output is zero.
u should then get similar values.

 

Yup. Just to add... convolution is often much more easily done in the frequency domain... if you can easily find the transforms (i.e. step function has a well known transform) all you have to do is transform x(t) and h(t) to get X(s) and H(S) (or X(jw) and H(jw) depending on which transform you use) and then multiply H(S) and X(S). Then take the inverse transform. And there's your answer. Its a property of convolution.

 

let start off with the basic:

how do you solve ∫(τ^2 + τ)u(τ)u(t-τ)dτ from -infinity to infinity?

I am stuck because there are 2 unit function

 

The easiest way to actually evaluate this integral (i.e. setting up the bounds) is by a graphical approach. Are you familiar with this at all?

 

Also, it may be helpful to think of h(t) as a shifted rect() function.

 

I am familiar with it. That is to change the bounds from the limits 0 to infinity.

 

Actually, graphical way of thinking is more easier than that.

As you know the unit step function makes lowest limit of integral 0. If you think the other positions, like SawabyPlus said, you can see that it is very easy to calculate.
If you're allowed to do it, then I suggest you to try that.

Best regards