The definition of the Laplace transform is \(\displaystyle{L}{\left({f}\right)}{\left({s}\right)}\:={\int_{{0}}^{{\infty}}}{e}^{{-{s}{t}}}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}{a}{s}{k}{\quad\text{and}\quad}{f}{\quad\text{or}\quad}{\text{sinh}{{t}}}{h}{e}{f}{o}{l}{l}{o}{w}\in{g}{h}{o}{l}{d}{s}:{P}{S}{K}{\sin{{h}}}{x}={\frac{{{1}}}{{{2}}}}{\left({e}^{{x}}-{e}^{{-{x}}}\right)}\). Now you put these two things together and compute

\(\displaystyle{\frac{{{1}}}{{{2}}}}{\int_{{0}}^{{\infty}}}{x}{e}^{{{4}{x}}}{e}^{{-{x}{s}}}{\left.{d}{x}\right.}-{\frac{{{1}}}{{{2}}}}{\int_{{0}}^{{\infty}}}{x}{e}^{{-{4}{x}}}{e}^{{-{x}{s}}}{\left.{d}{x}\right.}\)

Hope this helps

\(\displaystyle{\frac{{{1}}}{{{2}}}}{\int_{{0}}^{{\infty}}}{x}{e}^{{{4}{x}}}{e}^{{-{x}{s}}}{\left.{d}{x}\right.}-{\frac{{{1}}}{{{2}}}}{\int_{{0}}^{{\infty}}}{x}{e}^{{-{4}{x}}}{e}^{{-{x}{s}}}{\left.{d}{x}\right.}\)

Hope this helps