Since we use an integral to calculate the Laplace Transform, you might expect to use derivatives to go the other direction. Counterintuitively, this is not the case. Calculating the inverse Laplace Transform involves fewer actual computations. Most of the time, you will use formulas and tables that were derived from calculating the Laplace Transform.
Recommended Books on Amazon (affiliate links)  

Notation
The notation we used for the Laplace Transform, looked like this. \[ F(s) = \mathcal{L}\{ f(t) \} \] Intuitively, the notation for the Inverse Laplace Transform is written in a logical manner. \[ f(t) = \mathcal{L}^{1} \{ F(s) \} \]
Rules
Since the Laplace Transform uses integration, which is a linear operation, so too is the inverse Laplace Transform. This means that \[ \mathcal{L}^{1} \{ F(s) + G(s) \} = \mathcal{L}^{1} \{ F(s) \} + \mathcal{L}^{1} \{ G(s) \} \] For the same reason, using the constant \(a\), this equation also holds. \[ \mathcal{L}^{1} \{ aF(s) \} = a \mathcal{L}^{1} \{ F(s) \} \]
Calculating the Inverse Laplace Transform
To actually calculate the inverse Laplace Transform, we use tables. However, you will need to make sure your
partial fraction expansion skills are sharp.
Okay, here is the table you need to use for calculating inverse Laplace Transforms. The practice problems column contain links to practice problems using this table.
Laplace Transforms  

\( f(t) \) 
\(\displaystyle{ F(s) }\) 

Basic Functions  
\( t^n, ~ n = 1, 2, 3, \ldots \) 
\(\displaystyle{ \frac{n!}{s^{n+1}} }\) 

\( e^{at} \) 
\(\displaystyle{ \frac{1}{sa} }\) 

\( \sin(\alpha t) \) 
\(\displaystyle{ \frac{\alpha}{s^2 + \alpha^2} }\) 

\( \cos(at) \) 
\(\displaystyle{ \frac{s}{s^2 + a^2} }\) 

\( \sinh(at) \) 
\(\displaystyle{ \frac{a}{s^2  a^2} }\) 

\( \cosh(at) \) 
\(\displaystyle{ \frac{s}{s^2  a^2} }\) 

Special Functions  
\( \delta(t) \) unit impulse 
\( 1 \) 

\( \delta(t\tau) \) shifted unit impulse 
\( e^{\tau s} \) 

\( u(t) \) unit step 
\(\displaystyle{ \frac{1}{s} }\) 

\( u(t\tau) \) shifted unit step 
\(\displaystyle{ \frac{1}{s} e^{\tau s} }\) 

Combined Functions  
\( e^{at}\sin(\alpha t) \) 
\(\displaystyle{ \frac{\alpha}{(sa)^2 + \alpha^2} }\) 

\( f(t)u(ta) \) 
\(\displaystyle{ e^{sa} \mathcal{L}\{ f(t+a) \} }\) 

\( t^n e^{at}, ~ n = 1, 2, 3, \ldots \) 
\(\displaystyle{ \frac{n!}{(sa)^{n+1}} \} }\) 

Derivatives and Integrals  
\( f'(t) \) 
\( sF(s)  f(0) \) 

\( f''(t) \) 
\( s^2F(s)  sf(0)  f'(0) \) 

\(\displaystyle{ f^{(n)}(t) }\) 
\(\displaystyle{ s^nF(s)  s^{n1}f(0)  }\) \(\displaystyle{ s^{n2}f'(0)  . . .  f^{(n1)}(0) }\) 

\( \int_0^t{ f(v)~dv }\) 
\(\displaystyle{ \frac{F(s)}{s} }\) 
Okay, so why do we need Laplace Transforms? Why are they useful? We use them to solve differential equations that cannot be solved otherwise, sometimes involving some special functions. These special functions also have a purpose. Some that you will run across are the unit step function, unit impulse function and the square wave.
Practice
Unless otherwise instructed, calculate the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \) using a table. Give your answer in exact, completely factored form.
Basic
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4} \right\} }\)
Final Answer 

\( t^3/6 \)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\( t^3/6 \)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{6s+3} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{6s+3} \right\} }\)
Final Answer 

\(\displaystyle{ \frac{e^{t/2}}{6} }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{6s+3} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \frac{e^{t/2}}{6} }\)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s+1}{s^2+2} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s+1}{s^2+2} \right\} }\)
Final Answer 

\(\displaystyle{ \cos(t\sqrt{2}) + \frac{\sin(t\sqrt{2})}{\sqrt{2}} }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s+1}{s^2+2} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \cos(t\sqrt{2}) + \frac{\sin(t\sqrt{2})}{\sqrt{2}} }\)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^2+2s} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^2+2s} \right\} }\)
Final Answer 

\(\displaystyle{ \frac{1}{2} ( 1e^{2t} ) }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^2+2s} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \frac{1}{2} ( 1e^{2t} ) }\)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s}{(s+2)^2} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s}{(s+2)^2} \right\} }\)
Final Answer 

\( e^{2t}  2te^{2t} \)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s}{(s+2)^2} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\( e^{2t}  2te^{2t} \)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{se^{\pi s/2}}{s^2+1} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{se^{\pi s/2}}{s^2+1} \right\} }\)
Final Answer 

\( \cos(t\pi/2)u(t\pi/2) \)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{se^{\pi s/2}}{s^2+1} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\( \cos(t\pi/2)u(t\pi/2) \)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s}{s^2+2s+2} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s}{s^2+2s+2} \right\} }\)
Final Answer 

\( e^{t}( \cos(t)  \sin(t) ) \)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s}{s^2+2s+2} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\( e^{t}( \cos(t)  \sin(t) ) \)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{(s+2)^5} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{(s+2)^5} \right\} }\)
Final Answer 

\(\displaystyle{ \frac{t^4 e^{2t}}{24} } \)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{(s+2)^5} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \frac{t^4 e^{2t}}{24} } \)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s+8}{s^2+4s+13} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s+8}{s^2+4s+13} \right\} }\)
Final Answer 

\( e^{2t}( \cos(3t) + 2\sin(3t) ) \)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s+8}{s^2+4s+13} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\( e^{2t}( \cos(3t) + 2\sin(3t) ) \)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4+5s^2+4} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4+5s^2+4} \right\} }\)
Final Answer 

\(\displaystyle{ \frac{1}{6} (2\sin(t)  \sin(2t)) }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4+5s^2+4} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \frac{1}{6} (2\sin(t)  \sin(2t)) }\)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4 e^{10s}} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4 e^{10s}} \right\} }\)
Final Answer 

\(\displaystyle{ \frac{(t10)^3}{6} u(t10) }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4 e^{10s}} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \frac{(t10)^3}{6} u(t10) }\)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \ln \left[ \frac{s^2+9}{s^2+1} \right] \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \ln \left[ \frac{s^2+9}{s^2+1} \right] \right\} }\)
Final Answer 

\(\displaystyle{ \frac{2}{t}( \cos(t) \cos(3t)) }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \ln \left[ \frac{s^2+9}{s^2+1} \right] \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \frac{2}{t}( \cos(t) \cos(3t)) }\)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^416} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^416} \right\} }\)
Final Answer 

\( \frac{1}{32}( e^{2t}  e^{2t} 2\sin(2t) ) \)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^416} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\( \frac{1}{32}( e^{2t}  e^{2t} 2\sin(2t) ) \)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s^3}{(s^416)^2} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s^3}{(s^416)^2} \right\} }\)
Hint 

Use the result from the previous problem to solve this one.
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{s^3}{(s^416)^2} \right\} }\)
Hint
Use the result from the previous problem to solve this one.
Solution
video by blackpenredpen 

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ F(s) = \frac{1}{s3}  \frac{16}{s^2+9} }\)
Problem Statement
For \(\displaystyle{ F(s) = \frac{1}{s3}  \frac{16}{s^2+9} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \).
Solution
video by PatrickJMT 

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ F(s) = \frac{s+3}{s^2+4s+13} }\)
Problem Statement
For \(\displaystyle{ F(s) = \frac{s+3}{s^2+4s+13} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \).
Solution
video by Krista King Math 

Log in to rate this practice problem and to see it's current rating. 

Intermediate
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^3 (s^2+1)} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^3 (s^2+1)} \right\} }\)
Final Answer 

\( \cos(t) + t^2/2  1 \)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^3 (s^2+1)} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\( \cos(t) + t^2/2  1 \)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \arctan(1/s) \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \arctan(1/s) \right\} }\)
Final Answer 

\(\displaystyle{ \frac{\sin(t)}{t} }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \arctan(1/s) \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \frac{\sin(t)}{t} }\)
Log in to rate this practice problem and to see it's current rating. 

Advanced
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{\sqrt{s}} + \frac{1}{\sqrt{e^s}} \right\} }\)
Problem Statement 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{\sqrt{s}} + \frac{1}{\sqrt{e^s}} \right\} }\)
Final Answer 

\(\displaystyle{ \frac{1}{\sqrt{\pi t}} + \delta(t1/2) }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{\sqrt{s}} + \frac{1}{\sqrt{e^s}} \right\} }\)
Solution
video by blackpenredpen 

Final Answer
\(\displaystyle{ \frac{1}{\sqrt{\pi t}} + \delta(t1/2) }\)
Log in to rate this practice problem and to see it's current rating. 

Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4+4s^2+4} \right\} }\)
Problem Statement
Evaluate \(\displaystyle{ \mathcal{L}^{1} \left\{ \frac{1}{s^4+4s^2+4} \right\} }\)
Solution
video by blackpenredpen 

Log in to rate this practice problem and to see it's current rating. 

Really UNDERSTAND Calculus
Log in to rate this page and to see it's current rating.
external links you may find helpful 

To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 
I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me. 

Support 17Calculus on Patreon 

