Correcting a Laplace Transform Problem

[jayanthd]

Homework Statement

Why I am getting wrong answer related to this Laplace Transforms problem?

According to the book "Higher Engineering Mathematics 6th edition by John O Bird" page no. 583 one should get

(e^{-st}/(s^{2} + a^{2}))(a sin at - s cos at)

Homework Equations

∫e^{-st}cos at dt

The Attempt at a Solution

u = e^{-st}

du = -se^{-st} dt

Let dv = cos at dt

v = (sin at)/ a

Integrating by parts

∫e^{-st}cos at dt =

(e^{-st} sin at / a) + (s/a)∫e^{-st}sin at dt

= (e^{-st} sin at / a) + (s/a)[(-e^{-st} cos at / a) - (s/a)∫e^{-st}cos at dt]

= (e^{-st} sin at / a) - (s/a^{2} )(e^{-st} cos at) - s^{2}/a^{2})∫e^{-st}cos at dt]

Rearranging

(1 + (s^{2}/a^{2}))∫e^{-st}cos at dt =

(e^{-st} sin at / a) - (s/a^{2})(e^{-st} cos at)

= (e^{-st}/a^{2})(a sin at - s cos at)

∫e^{-st}cos at dt =

(a^{2}/(a^{2} + s^{2}))((e^{-st}/a^{2})(a sin at - s cos at))

= ((e^{-st}/ (s^{2} + a^{2}))(a sin at - s cos at)

 

[Ray Vickson]

Homework Statement

Why I am getting wrong answer related to this Laplace Transforms problem?

According to the book "Higher Engineering Mathematics 6th edition by John O Bird" page no. 583 one should get

(e^{-st}/(s^{2} + a^{2}))(a sin at - s cos at)

Homework Equations

∫e^{-st}cos at dt

The Attempt at a Solution

u = e^{-st}

du = -se^{-st} dt

Let dv = cos at dt

v = (sin at)/ a

Integrating by parts

∫e^{-st}cos at dt =

(e^{-st} sin at / a) + (s/a)∫e^{-st}sin at dt

= (e^{-st} sin at / a) + (s/a)[(-e^{-st} cos at / a) - (s/a)∫e^{-st}cos at dt]

= (e^{-st} sin at / a) - (s/a^{2} )(e^{-st} cos at) - s^{2}/a^{2})∫e^{-st}cos at dt]

Rearranging

(1 + (s^{2}/a^{2}))∫e^{-st}cos at dt =

(e^{-st} sin at / a) - (s/a^{2})(e^{-st} cos at)

= (e^{-st}/a^{2})(a sin at - s cos at)

∫e^{-st}cos at dt =

(a^{2}/(a^{2} + s^{2}))((e^{-st}/a^{2})(a sin at - s cos at))

= ((e^{-st}/ (s^{2} + a^{2}))(a sin at - s cos at)

There is no way you could get the answer that you claim the book obtains: the Laplace transform of a function f(t) will not have a "t" in it, since t has be "integrated out". I hope you realize that you need to take a limit!

 

[Chestermiller]

The integration limits on the LT integration are from 0 to ∞. Try these limits and see what you get. It's a definite integral, not an indefinite integral.

 

[jayanthd]

Applying limits on the last step was not a problem. I was getting wrong answer that is there was a mistake in sign at one place where Integration by parts is done second time. After fixing it I got the right answer which I have modified in post 1. After applying the limits I got the right answer.

e^{-s\infty} becomes 0

and

e^{-s * 0} becomes 1

sin 0 = 0

cos 0 = 1

First term of the equation after applying infinity becomes 0. Remaining is applying 0 as limit but a - sign appears before the equation.

- [ e^{-s * 0} / (s^{2} + a^{2})(a sin a (0) - s cos a (0))

= - (1 / (s^{2} + a^{2})) ( - s)

= (s / (s^{2} + a^{2}))

Am I right?