# Don't understand this application of the transform of an integral

1. Oct 2, 2013

### Turion

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

$$L\{ 1*{ t }^{ 3 }\}$$

2. Relevant equations

$$L\{ \int _{ 0 }^{ t }{ f(\tau )d\tau } \} =\frac { L\{ f(t)\} }{ s }$$

3. The attempt at a solution

$$L\{ 1*{ t }^{ 3 }\} \\ =L\{ \int _{ 0 }^{ t }{ { (t-\tau ) }^{ 3 }d\tau } \} \\ =\frac { L\{ { t }^{ 3 }\} }{ s } \\ =\frac { 6 }{ { s }^{ 5 } }$$

The function in the integral is a function of t and tau so how was the identity under "relevant equations" applied?

2. Oct 2, 2013

### szynkasz

Yes, you have $\int_0^tf(\tau,t)\,d\tau$ in the integral instead of $\int_0^tf(\tau)\,d\tau$ and the formula for integral doesn't work. This is convolution so:

$L\{1*t^3\}=L\{1\}\cdot L\{t^3\}$

3. Oct 2, 2013

### vela

Staff Emeritus
Use the substitution $u=t-\tau$ on the integral to convert it to a more familiar form.

4. Oct 3, 2013

### Turion

@szynkasz: Thanks. I solved it.

Any suggestions?

5. Oct 3, 2013

### vela

Staff Emeritus
Remember that $u$ is a dummy variable.

6. Oct 3, 2013

### Turion

Yeah, I substituted it back in the last step so there is no more u. I'm not sure how that would help me.

7. Oct 3, 2013

### vela

Staff Emeritus
I don't think you understood my point. In the integral $\int_0^t u^3\,du$, it doesn't matter what letter we use for the variable of integration. It's a dummy variable.

http://mathworld.wolfram.com/DummyVariable.html

If you evaluated the integral, what variable is it a function of?

8. Oct 3, 2013

### Turion

It's a function of u which is the same thing as saying it's a function of t - tau.

9. Oct 3, 2013

### vela

Staff Emeritus
Really? Did you actually calculate $\int_0^t u^3\,du$?

10. Oct 3, 2013

### Turion

No, I didn't. The textbook says to apply the identity under "2. Relevant equations" first.

11. Oct 3, 2013

### vela

Staff Emeritus
You wrote
What variable is $f$ a function of on the left-hand side? What variable is $f$ a function of on the right-hand side? Are they the same? Why not?

12. Oct 3, 2013

### Turion

On the left hand side, f is a function of tau. On the right hand side, f is a function of t. They are not the same. I'm not sure why they're not the same but I can dig up the proof for it I guess. I think they're not the same because we're doing a transformation.

Last edited: Oct 3, 2013
13. Oct 3, 2013

### Turion

I found the proof:

Honestly, that proof (as most proofs) went right over my head.