List of top Mathematics Questions on Calculus

If \( I = \int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}} \), then prove that \( I = \frac{\pi}{12} \).
Find the minimum value of the objective function \( Z = -50x + 20y \) by graphical method under the following constraints :
\( 2x - y \geq -5 \)
\( 3x + y \geq 3 \)
\( 2x - 3y \leq 12 \)
\( x \geq 0, y \geq 0 \)
Without cover a box is formed by 6 m x 16 m rectangular steel sheet on cutting the squares of length x m from its each corner. Then find the maximum volume of the box.
Prove that \( \sin^{-1} x = \tan^{-1} [x / \sqrt{1-x^2}] \).
Prove that \(\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\tan x}} = \frac{\pi}{12}\).
If \( \int \log x \, dx = x \log x + k(x) + c \) then
Find the area enclosed by the curve \( y = x^2 \) and the line \( y = 16 \).
A car is started from a point P at time t = 0 and is stopped at the point Q. The distance x metre covered by the car in t second is given by \( x = t^2(2 - \frac{t}{3}) \). Find the time required by the car to reach at point Q and also find the distance between P and Q.
Minimize Z = 3x + 2y by graphical method under the following constraints: \(x + 2y \leq 10\), \(3x + y \leq 15\), \(x \geq 0\), \(y \geq 0\).
Differentiate the function \(x^{\cos x}\) with respect to x.
A box is formed by a 3 m x 8 m rectangular steel-sheet on cutting the squares of length x m from its each corner to form the box without cover. Then find the maximum volume of the box so formed.
Find the area of the region bounded by the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Prove that the semi-vertical angle of the cone of given slant height and maximum volume is tan\(^{-1}\) \(\sqrt{2}\).
Sand is falling from a pipe at the rate of 12 cm\(^3\)/second. The falling sand forms such a cone on the ground that its height is always one-sixth of the radius of its base. At which rate is the height of the cone formed by sand increasing while its height is 4 cm?
Maximize \(Z = x + 2y\) by graphical method under the constraints \(x+y \le 1, -x+y \le 0, x \ge 0, y \ge 0\).
Evaluate : \(\int \frac{dx}{\sqrt{x^2 - a^2}}\).
Evaluate: \( \int \text{cosec}\,x(\text{cosec}\,x + \cot x) \,dx \).
Test whether the function defined by \( f(x) = x^2 - \sin(x) + 5 \) is continuous at \( x = \pi \).
Differentiate \(x^{\sin x}\) with respect to \(x\), while \(x>0\).
Find the area of the region bounded by the ellipse \(\frac{x^2}{9^2} + \frac{y^2}{4^2} = 1\).
Minimize Z = 200x + 500y by graphical method subject to the following constraints:
x + 2y \(\ge\) 10, 3x + 4y \(\le\) 24, x \(\ge\) 0, y \(\ge\) 0.
Prove that the semi-vertical angle of a right circular cone of given surface and maximum volume is \(\sin^{-1}(1/3)\).
Evaluate: \(\int \sqrt{3 - 2x - x^2} dx\).
If \(y = x^{x^{x^{\dots \text{to infinity}}}}\), prove that \(x \frac{dy}{dx} = \frac{y^2}{1 - y \log x}\).
Show that the function \(f(x) = \log \sin x\) is increasing in the interval \((0, \frac{\pi}{2})\) and decreasing in \((\frac{\pi}{2}, \pi)\).