How to use the Laplace Transform Table and Solver
- 1. Pick a functionChoose f(t) such as eᵃᵗ sin(bt), tⁿ or a unit step.
- 2. Enter the constantsType a, b or n. Add a value of s if you want a numerical check.
- 3. Read F(s)See the formula with your numbers, where it is valid, and the integral check.
Definition and the standard table
The Laplace transform of f(t) is F(s) = ∫₀^∞ e⁻ˢᵗ f(t) dt. It turns differential equations into algebra, which is why it is part of every engineering maths and control systems syllabus. The table in the result lists the 14 pairs you use most: 1, tⁿ, eᵃᵗ, sin bt, cos bt, sinh bt, cosh bt, tⁿeᵃᵗ, eᵃᵗ sin bt, eᵃᵗ cos bt, t sin bt, t cos bt, the unit step u(t − a) and the impulse δ(t − a).
Each pair holds only for s in its region of convergence, for example s > a for eᵃᵗ.
Worked example
Find L{e⁻ᵗ sin 2t}. Here a = −1 and b = 2. The first shifting theorem says L{eᵃᵗ f(t)} = F(s − a), and L{sin 2t} = 2/(s² + 4). Replace s with s + 1: F(s) = 2/((s + 1)² + 4), valid for s > −1.
At s = 2 this gives 2/(9 + 4) = 0.153846. The tool also integrates e⁻²ᵗ · e⁻ᵗ sin 2t from 0 to a large t with Simpson's rule and gets the same 0.153846, which confirms the table entry.
Properties to remember
Linearity: L{af + bg} = aF + bG, so L{3 sin 2t + 5} = 6/(s² + 4) + 5/s. Derivative: L{f′(t)} = sF(s) − f(0). Multiplying by t: L{t f(t)} = −dF/ds, which is where t sin bt comes from. Second shifting: L{f(t − a)u(t − a)} = e⁻ᵃˢF(s). For numerical integration of your own functions, use the Simpson's rule calculator.
Frequently asked questions
Can it find the inverse Laplace transform?
Use the table backwards: match your F(s) to a row, then read f(t). Split F(s) into partial fractions first if needed.
Why is there no numerical check for δ(t − a)?
The impulse is not an ordinary function, so it cannot be integrated with Simpson's rule. Its transform e⁻ᵃˢ comes from its sifting property.
What does 'valid for s > a' mean?
The integral only converges for those values of s. Outside that range the transform does not exist.
Is this tool free?
Yes. You enter your name and WhatsApp number once when you see your result, and other Uncodemy tools then open without the form.