List of top Mathematics Questions on Calculus

If the sum of the three numbers which are in A.P. is \(27\) and the product of the first and last is \(77\), then the numbers are

The area of the region bounded by the curves \[ y=2^x,\qquad y^2=4x \] and the lines \[ x=\frac12,\qquad x=1 \] is

Evaluate \[ \int_{0}^{2}x^{3}(4-x^{2})^{\frac52}\,dx \]

Evaluate \[ \int \frac{dx}{10\cos^2x+\cos2x+5} \]

If \([x]\) denotes the greatest integer function, then \[ \int_{0}^{5}[x-2]\,dx= \]

The general solution of the differential equation \[ x^{3}\,dx-xy^{2}\,dy+y^{3}\,dx=0 \] is

The differential equation corresponding to the family of curves \[ y=\log_e(ax+3), \] where \(a\) is an arbitrary constant, is

The local minimum value of the function \[ f(x)=2x-3\tan^{-1}x \] is

If \[ \int_{x}^{2x}\left(1+(\log 2)\log x\right)^{x}\,dx=f(x)+c \] and \[ f(1)=0, \] then \(f(e)=\)

If \[ f(x)=x^{3}-10x^{2}+31x-30 \] is a real valued function, then the number of values of \(c\), as stated in Rolle's theorem, that lie in the interval \((2,5)\) is

If \[ \int\frac{e^x+3}{\sqrt{e^x+4}}\,dx = 2\sqrt{e^x+4} +\frac32 \log\left| \frac{f(x)-2}{f(x)+2} \right| +c \] and \[ f(0)=\sqrt5, \] then \[ f(\log_e5)= \]

Evaluate \[ \int \frac{\tan 2x}{\cos^4 x}\,dx \]

If \[ f(x)=xe^{x^{2}-2x-3} \] is a real valued function, then \(f\) is

If \[ f(x)=\left(\log x\right)^{\sin x},\qquad x>e, \] then \(f'(\pi)=\)

If \[ y=\sin^{-1}\!\left(\frac{1-\cos2x}{1+\sin^{4}x}\right), \] then \[ \frac{dy}{dx}= \]

If \[ y=\left(x+\sqrt{x^{2}+1}\right)^{5}, \] then \(25y=\)

If the parabola \[ y^{2}=9x \] cuts the ellipse \[ \frac{x^{2}}{9}+\frac{y^{2}}{b}=1 \] orthogonally, then the length of the latus rectum of the given parabola is

Approximate volume of a cone whose semi-vertical angle is \[ \tan^{-1}(3) \] and base radius is \(15.001\) is

If \[ y=\tan^{-1}\!\left(\frac{\sqrt{1+9x^{2}}-1}{3x}\right), \] then \[ \left(\frac{dy}{dx}\right)_{x=\frac1{\sqrt3}} = \]

If the function \[ f(x)= \begin{cases} \dfrac{\sin x-\tan x}{x^{3}}, & -\dfrac{\pi}{2}[1ex] a, & x=0,[1ex] \dfrac{\sin (b-3)x+\sin bx}{\sin x}, & 0<x<\dfrac{\pi}{2}, \end{cases} \] is continuous in \[ \left(-\dfrac{\pi}{2},\,\dfrac{\pi}{2}\right), \] then \(a+2b=\)

If \(a\) and \(b\) are non-negative real numbers and \[ \lim_{x\to0}\frac{e^{ax}-\cos bx}{1-\cos x}=4, \] then \[ \lim_{x\to a}\frac{\sin(bx-ab)}{x-a} = \]

Evaluate \[ \lim_{x\to0}\frac{e^{\tan x}-e^{\sin x}}{(1-\cos x)\tan x} \]

Evaluate \[ \lim_{x\to0}\frac{e^{\tan x}-e^{\sin x}}{(1-\cos x)\tan x} \]

The value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{32 \cos^4 x}{1 + e^{\sin x}} dx$ is: