## Problem 03 | Evaluation of Integrals

**Problem 03**

Find the value of $\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt$

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**Problem 03**

Find the value of $\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt$

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**Problem 03**

Find the Laplace transform of $f(t) = t^2 \cos 3t$.

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**Problem 02**

Find the Laplace transform of $f(t) = t \sin 3t$.

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**Problem 01**

Find the Laplace transform of $f(t) = t \cos 2t$.

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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*.

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