How to use the Inverse Laplace Transform Calculator
- 1. Enter F(s)Type the numerator and the denominator, factored like (s − 1)(s² + 2s + 5) or expanded.
- 2. Press Find f(t)The tool factors the denominator, finds the constants and maps each term to the time domain.
- 3. Check the answerIt integrates e^(−st) f(t) numerically and compares it with F(s) at one value of s.
How the inverse Laplace transform is found
Most exam questions give F(s) as a ratio of two polynomials. The method is always the same: split F(s) into partial fractions, then use a standard pair for each piece. A linear factor gives an exponential, L⁻¹{1/(s − a)} = e^(at). A repeated factor gives a power of t, L⁻¹{1/(s − a)ⁿ} = tⁿ⁻¹e^(at)/(n − 1)!. A quadratic that does not factor is written as (s + α)² + β², and the first shifting theorem gives e^(−αt) cos βt and e^(−αt) sin βt.
If the numerator's degree is not lower than the denominator's, the tool divides first; the quotient gives impulses δ(t). Squared quadratic factors use L⁻¹{s/(s² + b²)²} = t sin bt/(2b) and L⁻¹{1/(s² + b²)²} = (sin bt − bt cos bt)/(2b³).
Worked example
Find L⁻¹{(5s + 3)/((s − 1)(s² + 2s + 5))}. Write it as A/(s − 1) + (Bs + C)/(s² + 2s + 5). Multiplying out, 5s + 3 = A(s² + 2s + 5) + (Bs + C)(s − 1). Comparing coefficients: A + B = 0, 2A − B + C = 5, 5A − C = 3, so A = 1, B = −1, C = 2.
Now s² + 2s + 5 = (s + 1)² + 4, and −s + 2 = −(s + 1) + 3. So the second term is −(s + 1)/((s + 1)² + 2²) + (3/2)·2/((s + 1)² + 2²). The answer is f(t) = e^t − e^(−t) cos 2t + (3/2) e^(−t) sin 2t, which is what the tool shows with its default input.
Tips for exams
Check each linear constant with the cover-up rule: put s = a in the rest of the fraction. Always complete the square before using the shifting theorem, and keep the factor β in the numerator for the sine term. To go the other way, use the Laplace transform table.
Frequently asked questions
Which denominators can it handle?
Any polynomial up to degree 12: linear factors, repeated linear factors, irreducible quadratics and squared quadratics. Write the denominator factored or expanded; integer cubics and quartics are factored for you.
What if the numerator degree is too high?
The tool divides first. The polynomial quotient becomes impulse terms δ(t), δ′(t) and so on, and the remainder is split into partial fractions.
How do I know the answer is right?
The tool computes ∫₀^∞ e^(−st) f(t) dt numerically at one value of s and compares it with F(s). A match means the f(t) shown is correct.
Why do I see decimals like 1.414214?
The denominator has irrational roots, for example s² − 2. The answer is still correct, just written with decimals.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.