10.4.30 Laplace transform

condition: Find (continuous on \( [0 ;+\infty)) \) function original if its image \( F(p) \) is equal to: 1) \( F(p)=\int_{p}^{+\infty} \frac{d q}{q\left(q^{2}+1\right)} \), 2) \( F(p)=\sum_{k=0}^{\infty} \frac{1}{p^{k+1}} \), 3) \( F(p)=\sum_{k=0}^{\infty} \frac{1}{(2 p)^{k+1}} \).