## Problem 04 | Evaluation of Integrals

**Problem 04**

Evaluate $\displaystyle \int_0^\infty \dfrac{e^{-t}\sin t}{t} ~ dt$.

- Read more about Problem 04 | Evaluation of Integrals
- Log in or register to post comments

**Problem 04**

Evaluate $\displaystyle \int_0^\infty \dfrac{e^{-t}\sin t}{t} ~ dt$.

- Read more about Problem 04 | Evaluation of Integrals
- Log in or register to post comments

**Problem 02**

Find the value of $\displaystyle \int_0^\infty \dfrac{\sin t ~dt}{t}$.

- Read more about Problem 02 | Evaluation of Integrals
- Log in or register to post comments

**Problem 01**

Evaluate $\displaystyle \int_0^\infty \dfrac{e^{-3t} - e^{-6t}}{t} ~ dt$

- Read more about Problem 01 | Evaluation of Integrals
- Log in or register to post comments

**Problem 04**

Find the Laplace transform of $f(t) = \dfrac{\cos 4t - \cos 5t}{t}$.

- Read more about Problem 04 | Division by t
- Log in or register to post comments

**Problem 03**

Find the Laplace transform of $f(t) = \dfrac{\sin^2 t}{t}$.

- Read more about Problem 03 | Division by t
- Log in or register to post comments

**Problem 02**

Find the Laplace transform of $f(t) = \dfrac{e^{4t} - e^{-3t}}{t}$.

- Read more about Problem 02 | Division by t
- Log in or register to post comments

**Problem 01**

Find the Laplace transform of $f(t) = \dfrac{\sin t}{t}$.

- Read more about Problem 01 | Division by t
- Log in or register to post comments

Let *f*(*t*) be a given function which is defined for *t* ≥ 0. If there exists a function *F*(*s*) so that

$\displaystyle F(s) = \int_0^\infty e^{-st} \, f(t) \, dt$,

then *F*(*s*) is called the **Laplace Transform** of *f*(*t*), and will be denoted by $\mathcal{L} \left\{f(t)\right\}$. Notice the integrator e^{-st} dt where *s* is a parameter which may be real or complex.

Thus,

$\mathcal{L} \left\{f(t)\right\} = F(s)$

The symbol $\mathcal{L}$ which transform *f*(*t*) into *F*(*s*) is called the *Laplace transform operator*.

- Read more about Laplace Transform
- Log in or register to post comments