The figure gives the electric potential $V$ as a function of distance through five regions on $x -$ axis. Which of the following is true for the electric field $E$ in these regions?

Join BYJU'S Learning Program

Let x(t) be signal with its Laplace transform X(s). If x(t) is defined as x(t)=e−tcostu(t). Another function y(t) is defined as y(t)=∫τ−∞x(τ)dτ. Then the Laplace transform of y(t) is

The correct option is C s+1s(s2+2s+2) Given, x(t)=e−tcostu(t) x(t)⟷LT X(s)=s+1(s+1)2+1=s+1s2+2s+2 y(t)=∫t−∞x(τ)dτ=X(s)s ∴Y(s)=s+1s(s2+2s+2)

$Let x(t) be signal with its Laplace transform X(s). If x(t) is defined as x(t) = e^{- t} cos t u (t) . Another function y(t) is defined as y (t) = \int_{- \infty}^{τ} x (τ) d τ$ .
$Then the Laplace transform of y(t) is$

The correct option is C $\frac{s + 1}{s (s^{2} + 2 s + 2)}$
$Given, x (t) = e^{- t} cos t u (t)$

$x (t) ⟷ L T$

$X (s) = \frac{s + 1}{(s + 1)^{2} + 1} = \frac{s + 1}{s^{2} + 2 s + 2}$

$y (t) = \int_{- \infty}^{t} x (τ) d τ = \frac{X (s)}{s}$

$∴ Y (s) = \frac{s + 1}{s (s^{2} + 2 s + 2)}$