Time Integration Property of Laplace Transform

Laplace Transform

The Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain or s-domain.

Mathematically, if x(t) is a time domain function, then its Laplace transform is defined as −

$$\mathrm{L\left[x(t)\right] \:=\:X(s) \:=\: \int_{-\infty}^{\infty} x(t) e^{-st} \: dt \quad \dotso\: (1)}$$

Integration in Time Domain Property of Laplace Transform

Statement - The time integration property of Laplace transform states that if

$$\mathrm{x(t) \: \overset{LT}\leftrightarrow\:X(s)}$$

Then

$$\mathrm{\int_{-\infty}^{t}\: x(\tau) \: d\tau\: \overset{LT}\leftrightarrow \:\frac{x(s)}{s} \:+\: \:\int_{-\infty}^{0}\: \frac{x(\tau)} {s}\: d\tau}$$

Proof

Consider a function y(t) as,

$$\mathrm{y(t) \:=\: \int_{-\infty}^{t}\: x(\tau) \: d\tau}$$

Taking differentiation on both sides with respect to time, we have,

$$\mathrm{\frac{d}{dt}\: y(t) \:=\: x(t) \quad \dotso\: (2)}$$

Also,

$$\mathrm{y(0^-) \:=\: \int_{-\infty}^{0}\: x(\tau) \: d\tau \quad \dotso\: (3)}$$

Taking the Laplace transform of equation (2), we get,

$$\mathrm{L \left[ \frac{d}{dt} \:y(t) \right] \:=\: L \left[ x(t) \right]}$$

$$\mathrm{\Rightarrow\: sY(s) \:-\: y(0^-) \:=\: X(s)}$$

$$\mathrm{\Rightarrow\: Y(s) \:=\: \frac{X(s)}{s} \:+\: \frac{y(0^-)}{s} \quad \dotso\: (4)}$$

From equations (3) and (4), we obtain,

$$\mathrm{Y(s) \:=\: \frac{X(s)}{s} \:+\: \int_{-\infty}^{0}\: x(\tau) s \: d\tau}$$

$$\mathrm{\therefore\: Y(s) \:=\: L\left[ \int_{-\infty}^{t}\: x(\tau) \: d\tau \right] \:=\: \frac{X(s)}{s} \:+\: \int_{-\infty}^{0}\: \frac{x(\tau)}{s} \: d\tau}$$

Or it can also be represented as,

$$\mathrm{\int_{-\infty}^{t}\: x(\tau) \: d\tau\: \overset{LT}\leftrightarrow \: \frac{X(s)}{s} \:+\: \int_{-\infty}^{0} \frac{x(\tau)}{s} \: d\tau}$$

Thus, it proves the integration in time domain property of the Laplace transform.