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`