integration

Consider a continuous-time signal x(t) = -t^2{u(t + 4) - u(t-4)} where u(t) is the continuous-time unit step function. Let δ(t) be the continuous-time unit impulse function. The va...

Let continuous-time signals $x_1(t)$ and $x_2(t)$ be $x_1(t) = \begin{cases} 1, & t \in [0,1] \\ 2-t, & t \in [1,2] \\ 0, & \text{otherwise} \end{cases}$ and $x_2(t) = \begin{cases...

Let continuous-time signals $x_1(t)$ and $x_2(t)$ be $x_1(t)=\left\{\begin{array}{cc}1, & t \in[0,1] \\ 2-t, & t \in[1,2] \\ 0, & \text { otherwise }\end{array}\right.$ and $x_2(t)...

Let $X(\omega)$ be the Fourier transform of the signal $x(t) = e^{-t^4} \cos t, \quad -\infty The value of the derivative of $X(\omega)$ at $\, \omega = 0$ is ______ (rounded off t...

Let the probability density function of a random variable x be given as f(x) = ae $$-$$2|x| The value of a is _________.

Let $f$ be a real-valued function of a real variable defined as $f(x)=x-[x]$, where $[x]$ denotes the largest integer less than or equal to $x$. The value of $\int_{0.25}^{1.25} f(...

Let the probability density function of a random variable $$X,$$ be given as: $$${f_x}\left( x \right) = {3 \over 2}{e^{ - 3x}}u\left( x \right) + a{e^{4x}}u\left( { - x} \right)$$...

A differential equation $$\,\,{{di} \over {dt}} - 0.21 = 0\,\,$$ is applicable over $$\,\, - 10 < t < 10.\,\,$$ If $$i(4)=10,$$ then $$i(-5)$$ is

A particle, starting from origin at $$t=0$$ $$s,$$ is traveling along $$x$$-axis with velocity $$v = {\pi \over 2}\cos \left( {{\pi \over 2}t} \right)m/s$$ At $$t=3$$ $$s,$$ the di...

Lifetime of an electric bulb is a random variable with density $$f\left( x \right) = k{x^2},$$ where $$x$$ is measured in years. If the minimum and maximum lifetimes of bulb are $$...

A continuous random variable $$X$$ has a probability density function $$f\left( x \right) = {e^{ - x}},0 < x < \infty .$$ Then $$P\left\{ {X > 1} \right\}$$ is

The input x(t) and output y(t) of a system are related as $$\int_{-\infty}^tx\left(\tau\right)\cos\left(3\tau\right)d\tau$$.The system is

Given two continuous time signals $$x\left(t\right)=e^{-t}$$ and $$y\left(t\right)=e^{-2t}$$ which exist for t > 0, the convolution z(t) = x(t)*y(t) is

The response h(t) of a linear time invariant system to an impulse $$\delta\left(t\right)$$, under initially relaxed condition is $$h\left(t\right)=e^{-t}\;+\;e^{-2t}$$. The respons...

The response $$h(t)$$ of a linear time invariant system to an impulse $$\delta \left( t \right),$$ under initially relaxed condition is $$h\left( t \right) = \,{e^{ - t}} + {e^{ -...

With $$K$$ as constant, the possible solution for the first order differential equation $${{dy} \over {dx}} = {e^{ - 3x}}$$ is

The expression $$V = \int\limits_0^H {\pi {R^2}{{\left( {1 - {h \over H}} \right)}^2}dh\,\,\,} $$ for the volume of a cone is equal to _________.

A voltage wafeform $$v\left( t \right) = 12\,{t^2}$$ is applied across a $$1$$ $$H$$ inductor for $$t \ge 0,$$ with initial current through it being zero. The current through the i...

The unit impulse response of a system is given as $$c\left( t \right) = - 4{e^{ - t}} + 6{e^{ - 2t}}.\,\,\,$$ The step response of the same system for $$\,t \ge 0$$ is equal to

The integration of $$\int {{\mathop{\rm logx}\nolimits} \,dx} $$ has the value