**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.

### 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) = e

^{t} + 4e

^{-t} + 2t

^{2} + 1

** Solution: **

Step 1: The given function is f(t) = e^{t} + 4e^{-t} + 2t^{2 }+ 1

Step 2:
Using the formula
We have

L[e^{t} + 4e^{-t} + 2t^{2} + 1] = L(e^{t}) + 4 L[e^{-t}] + 2 L[t^{2}] + 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[e^{t} + 4e^{-t} + 2t^{2} + 1]
= $\frac{1}{s - 1}$ + $\frac{4}{s + 1}$ + $\frac{4}{s^{3}}$ + $\frac{1}{s}$