Skip to main content
Search IntMath
Close

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

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`

Answer

In this example, g(t) = cos at and from the Table of Laplace Transforms, we have:

`G(s)= Lap{cosat}` `=s/((s^2+a^2))`

Now, applying the first rule above, we have:

`Lap{int_0^tcosat\ dt}=1/sxxs/(s^2+a^2)`

`=1/(s^2+a^2)`

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

Answer

This is similar to example (a). We find the transform of the function g(t) = eatcos bt, then divide by s, since we are finding the Laplace transform of the integral of g(t) evaluated from 0 to t.

`Lap{int_0^te^(at)cosbt\ dt}=1/sxx(s-a)/((s-a)^2+b^2)`

`=(s-a)/(s((s-a)^2+b^2)`

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

Answer

This follows the same process as examples (a) and (b).

Find the Laplace transform of the function `g(t)=te^(-3t)` then divide by s.

`Lap{int_0^t te^(-3t)dt}=1/sxx1/((s+3)^2)`

`=1/(s(s+3)^2)`

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

Answer

Recall from the Double Angle Formula that

`sin 2α = 2\ sin α\ cos α`

We can use this to re-express our integrand (the part we are integrating):

`sin at\ cos at=1/2 sin 2at`

So the Laplace Transform of the integral becomes:

`Lap{int_0^t\ sin at\ cos at\ dt}=1/2 Lap{int_0^t\ sin 2at\ dt}`

`=1/2(2a)/(s(s^2+4a^2))`

`=a/(s(s^2+4a^2))`