Let f be the function given defined for variable function t between 0 and $\infty$ and s be positive transforming variable.

In general Laplace Transform formula is given by

Standard Laplace Transform Formulas are given below which may be helpful for you to solve instantly instead of making your problem lengthy.

Laplace transform Calculator calculates the laplace transform of the given function. It is a easy tool to get your answer instantly.

## Laplace Transform Problems

Steps to solve Laplace transform

Step 1: Read the problem and observe the function given.

Step 2 : Use the laplace transform formula for the function given
F(s) = $\int_{0}^{\infty}$ f(t) e-st dt
or use standard Laplace transform formulas given above for the function given.

Step 3 : Simplify the algebraic expression from which you get the final answer.

Below are given some problems based on Laplace transform which may be useful for you.

### Solved Examples

Question 1: Find the Laplace transform of sin 5t ?
Solution:

Step 1: The given function is sin 5t
Step 2: Using the formula L[sin at] = $\frac{a}{s^{2} + a^{2}}$
We have L[sin 5t] = $\frac{5}{s^{2} + 5^{2}}$
Step 3: On simplification we get L[sin 5t] = $\frac{5}{s^{2} + 25}$

Question 2: Find the Laplace transform for the function:
f(t) = et + 4e-t + 2t2 + 1
Solution:

Step 1: The given function is f(t) = et + 4e-t + 2t2 + 1
Step 2: Using the formula We have
L[et + 4e-t + 2t2 + 1] = L(et) + 4 L[e-t] + 2 L[t2] + L[1]
= $\frac{1}{s - 1}$ + 4 $\frac{1}{s + 1}$ + 2 $\frac{2}{s^{2+1}}$ + $\frac{1}{s}$

Step 3: On simplification we get L[et + 4e-t + 2t2 + 1] = $\frac{1}{s - 1}$$\frac{4}{s + 1}$$\frac{4}{s^{3}}$ + $\frac{1}{s}$

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