On the evaluation of $L^{-1}(\frac{1}{p^3+1})$

Question

Evaluate $$L^{-1}(\frac{1}{p^3+1})$$

Any clue on this? i am absolutely getting no idea. Pls provide me some pointers related to this.

Answer 1

Hint:

We have: $$L^{-1} \left[ \frac1{1+p^3} \right] = L^{-1} \left[\frac1{(1+p)(1-p+p^2)} \right] $$

Using partial fraction decomposition, we have: $$\frac13\left[ L^{-1} \left(\frac{1}{1+p}\right) - L^{-1} \left[ \frac{p-\frac12} {(p-\frac12)^2 + \left(\frac{\sqrt 3}{2} \right)^2}\right] + \frac32 L^{-1} \left(\frac1{(p-\frac12)^2 + \left(\frac{\sqrt 3}{2} \right)^2} \right) \right] $$

Can you take it from here?