Question

The The Laplace Transform of a function , which is defined for all , is denoted by and is defined by the improper integral , as long as it converges. Laplace Transform is very useful in physics and engineering for solving certain linear ordinary differential equations. (Hint: think of as a fixed constant) 1. Find (hint: remember integration by parts)

Answer 1

Answer:

a. L{t} = 1/s² b. L{1} = 1/s

Step-by-step explanation:

Here is the complete question

The The Laplace Transform of a function ft), which is defined for all t2 0, is denoted by Lf(t)) and is defined by the improper integral Lf))s)J" e-st . f(C)dt, as long as it converges. Laplace Transform is very useful in physics and engineering for solving certain linear ordinary differential equations. (Hint: think of s as a fixed constant) 1. Find Lft) (hint: remember integration by parts) A. None of these. B. O C. D. 1 E. F. -s2 2. Find L(1) A. 1 B. None of these. C. 1 D.-s E. 0

Solution

a. L{t}

L{t} = ∫₀⁰⁰$e^{-st}t$

Integrating by parts  ∫udv/dt = uv - ∫vdu/dt where u = t and dv/dt = $e^{-st}$ and v = $\frac{e^{-st}}{-s}$ and du/dt = dt/dt = 1

So, ∫₀⁰⁰udv/dt = uv - ∫₀⁰⁰vdu/dt w

So,  ∫₀⁰⁰$e^{-st}t$ =  [$\frac{te^{-st}}{-s}$]₀⁰⁰ -  ∫₀⁰⁰ $\frac{e^{-st}}{-s}$

∫₀⁰⁰$e^{-st}t$ =  [$\frac{te^{-st}}{-s}$]₀⁰⁰ -  ∫₀⁰⁰ $\frac{e^{-st}}{-s}$

= -1/s(∞exp(-∞s) - 0 × exp(-0s)) + $\frac{1}{s}$ [$\frac{e^{-st} }{-s}$]₀⁰⁰

= -1/s[(∞exp(-∞) - 0 × exp(0)] - 1/s²[exp(-∞s) - exp(-0s)]

= -1/s[(∞ × 0 - 0 × 1] - 1/s²[exp(-∞) - exp(-0)]

= -1/s[(0 - 0] - 1/s²[0 - 1]

= -1/s[(0] - 1/s²[- 1]

= 0 + 1/s²

= 1/s²

L{t} = 1/s²

b. L{1}

L{1} = ∫₀⁰⁰$e^{-st}1$

= [$\frac{e^{-st} }{-s}$]₀⁰⁰

= -1/s[exp(-∞s) - exp(-0s)]

= -1/s[exp(-∞) - exp(-0)]

= -1/s[0 - 1]

= -1/s(-1)

= 1/s

L{1} = 1/s