Laplace Transform of t | Laplace of t

The Laplace transform of t is equal to L{t} = 1/s². Here we will find the Laplace of t by the definition of Laplace transforms.

The formula of the Laplace of t is given as follows:

L{t} = $\dfrac{1}{s^2}$.

Find Laplace Transform of t

Recall, the definition of the Laplace transform. According to this, the Laplace of f(t) is given by the integral

L{f(t)} = $\int_0^\infty$ e^-st f(t) dt

Let f(t) = t.

∴ L{t} = $\int_0^\infty$ te^-st dt

= $\Big[t\int e^{-st} dt – \int \{\frac{d}{dt}(t) \int e^{-st} dt\} dt \Big]_0^\infty$, using the integration by parts.

= $\Big[ t \cdot \dfrac{e^{-st}}{-s} – \int \dfrac{e^{-st}}{-s} dt \Big]_0^\infty$

= $\Big[ \dfrac{te^{-st}}{-s} – \dfrac{e^{-st}}{s^2} \Big]_0^\infty$

= $\lim\limits_{t \to \infty}\Big( \dfrac{te^{-st}}{-s} – \dfrac{e^{-st}}{s^2} \Big)$ $-\big( 0- \dfrac{1}{s^2}\big)$

= $0-0 + \dfrac{1}{s^2}$ as lim_t→∞ e^-st=0.

= $\dfrac{1}{s^2}$.

So the Laplace transform of t is 1/s².

L{t} = 1/s2.

Related Topics:

Laplace transform of t²

Laplace transform of 2

Laplace transform of sint

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