1. Find the Laplace transform for $\displaystyle f\left(t\right)=t\,e^ {- t }$

    $\displaystyle {{1}\over{\left(s-1\right)^2}}$ $\displaystyle {{2}\over{\left(s-1\right)^3}}$ $\displaystyle {{2}\over{\left(s+1\right)^3}}$ $\displaystyle {{1}\over{\left(s+1\right)^2}}$ $\displaystyle {{1}\over{s+1}}$

  2. Find the Laplace transform for $\displaystyle f\left(t\right)=e^ {- 2\,t }\,\cos \left(3\,t\right)$

    $\displaystyle {{2}\over{s^2+4\,s}}$ $\displaystyle {{3}\over{s^2+4\,s-5}}$ $\displaystyle {{s+2}\over{s^2+4\,s+8}}$ $\displaystyle {{s+2}\over{s^2+4\,s+13}}$

  3. Find the Laplace transform for $\displaystyle f\left(t\right)=e^{t}\,\cosh \left(3\,t\right)$

    $\displaystyle {{s}\over{s^2-2\,s-8}}-{{1}\over{s^2-2\,s-8}}$ $\displaystyle {{s}\over{s^2-2\,s+10}}-{{1}\over{s^2-2\,s+10}}$ $\displaystyle {{2\,s^2}\over{s^4+18\,s^2+81}}-{{1}\over{s^2+9}}$ $\displaystyle {{2\,s^2}\over{s^4-18\,s^2+81}}-{{1}\over{s^2-9}}$

  4. Find the Laplace transform for $\displaystyle f\left(t\right)=\int_{0}^{t}{e^{u}\,\sinh u\;du}$

    $\displaystyle {{s}\over{s^3-2\,s^2}}-{{1}\over{s^3-2\,s^2}}$ $\displaystyle {{1}\over{s^3-2\,s^2}}$ $\displaystyle {{2}\over{s^4-2\,s^2+1}}$ $\displaystyle {{2\,s}\over{s^4-2\,s^2+1}}-{{1}\over{s^3-s}}$

  5. Find the inverse transform for $\displaystyle F\left(s\right)={{s}\over{\left(s+1\right)^3}}$

    $\displaystyle t\,e^{t}$ $\displaystyle {{t^2\,e^{t}}\over{2}}+t\,e^{t}$ $\displaystyle t\,e^ {- t }$ $\displaystyle t\,e^ {- t }-{{t^2\,e^ {- t }}\over{2}}$

  6. Find the inverse transform for $\displaystyle F\left(s\right)={{s}\over{s^2-4\,s+13}}$

    $\displaystyle {{e^{2\,t}\,\sin \left(3\,t\right)}\over{3}}$ $\displaystyle {{e^{2\,t}\,\sin \left(2\,t\right)}\over{2}}$ $\displaystyle e^{2\,t}\,\left(\sin \left(2\,t\right)+\cos \left(2\,t\right) \right)$ $\displaystyle e^{2\,t}\,\left({{2\,\sin \left(3\,t\right)}\over{3}}+\cos \left(3 \,t\right)\right)$

  7. Find the inverse transform for $\displaystyle F\left(s\right)=-{{1}\over{2\,\left(s^2+4\right)}}$

    $\displaystyle {{t\,\sin \left(2\,t\right)}\over{4}}$ $\displaystyle -{{\sin \left(2\,t\right)}\over{4}}$ $\displaystyle -{{\cos \left(2\,t\right)}\over{2}}$ $\displaystyle {{t\,\cos \left(2\,t\right)}\over{2}}$

  8. Find the inverse transform for $\displaystyle F\left(s\right)={{1}\over{\left(s^2+1\right)^2}}$

    $\displaystyle \sin t$ $\displaystyle {{\sin t}\over{2}}-{{t\,\cos t}\over{2}}$ $\displaystyle \cos t$ $\displaystyle {{t\,\sin t}\over{2}}$


Department of Mathematics
Last modified: 2026-04-21