integration

Integration of $\ln (x)$ with $x$ i.e. $$ \int \ln (x) d x= $$__________

The second-order differential equation in an unknown function $$u : u(x, y)$$ is defined as $$\frac{\partial^2 u}{\partial x^2}= 2$$ Assuming $$g : g(x)$$, $$f : f(y)$$, and $$h :...

$\int \left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots\right) dx$ is equal to

$$\int {\left( {x - {{{x^2}} \over 2} + {{{x^3}} \over 3} - {{{x^4}} \over 4} + ....} \right)dx} $$ is equal to :

The value (round off to one decimal place) of $\int_{-1}^{1} x e^{|x|} dx$ is

Consider the hemi-spherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m³) when the depth of water at the centre of the tank i...

A particle of mass 2 kg is travelling at a velocity of 1.5 m/s. A force f (t) = 3t^2 (in N) is applied to it in the direction of motion for a duration of 2 seconds, where t denotes...

Let $$x$$ be a continuous variable defined over the interval $$\left( { - \infty ,\infty } \right)$$, and $$f\left( x \right) = {e^{ - x - {e^{ - x}}}}.$$ The integral $$g\left( x...

For the function $$\,f\left( x \right) = a + bx,0 \le x \le 1,\,\,$$ to be a valid probability density function, which one of the following statements is correct?

Consider the following second order linear differential equation $${{{d^2}y} \over {d{x^2}}} = - 12{x^2} + 24x - 20$$ The boundary conditions are: at $$x=0, y=5$$ and at $$x=2, y=2...

If $$\left\{ x \right\}$$ is a continuous, real valued random variable defined over the interval $$\left( { - \infty ,\,\, \pm \infty } \right)$$ and its occurrence is defined by t...

Find the value of $$\lambda $$ such that the function $$f(x)$$ is a valid probability density function ________.

Given two continuous time signals $$x\left( t \right) = {e^{ - t}}$$ and $$y\left( t \right) = {e^{ - 2t}}$$ which exists for $$t>0$$ then the convolution $$z\left( t \right) = x\l...

A parabolic cable is held between two supports at the same level. The horizontal span between the supports is $$L.$$ The sag at the mid-span is $$h.$$ The equation of the parabola...

The Laplace transform of the following function is $$$f\left( t \right) = \left\{ {\matrix{ {\sin t} & {for\,\,0 \le t \le \pi } \cr 0 & {for\,\,t > \pi } \cr } } \right.$$$

The value of the following definite integral in $$\int\limits_{{\raise0.5ex\hbox{$\scriptstyle { - \pi }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}^{{\raise0....

The solution for the following differential equation with boundary conditions $$y(0)=2$$ and $$\,\,{y^1}\left( 1 \right) = - 3$$ is where $${{{d^2}y} \over {d{x^2}}} = 3x - 2$$

The Laplace transform of the function $$\eqalign{ & f\left( t \right) = k,\,0 < t < c \cr & \,\,\,\,\,\,\,\,\, = 0,\,c < t < \infty ,\,\, \cr} $$ is

If $$c$$ is a constant, then the solution of $${{dy} \over {dx}} = 1 + {y^2}$$ is