# 6. Laplace Transforms of Integrals

We first saw the following properties in the Table of Laplace Transforms.

1. If `G(s)= Lap{g(t)}`, then `Lap{int_0^tg(t)dt}=(G(s))/s`.

2. For the general integral, if

`[intg(t)dt]_(t=0)`

is the value of the integral when `t=0`, then:

`Lap{intg(t)dt}` `=(G(s))/s+1/s[intg(t)dt]_(t=0)`

### Examples

Use the above information and the Table of Laplace Transforms to find the Laplace transforms of the following integrals:

(a) `int_0^tcos\ at\ dt`

(b) `int_0^te^(at)cos\ bt\ dt`

(c) `int_0^t te^(-3t) dt`

(d) `int_0^tsin\ at\ cos\ at\ dt`

### Search IntMath, blog and Forum

### Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Go to: Online algebra solver

### Calculus Lessons on DVD

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!