List of top Mathematics Questions on Calculus asked in UP Board XII

Solve the integral \(\int \frac{\sec^2 2\theta}{(\cot \theta - \tan \theta)^2} \, d\theta\).
Find the area bounded by the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).
Solve: \(\int \frac{3x - 5}{x^3 - x^2 - x + 1} dx\)
Prove that the radius of the right circular cylinder of maximum curved surface inscribed in a cone is half of the radius of the cone.
Find the interval in which the given function \( f(x) = x^2 - 4x + 6 \) is (i) Increasing (ii) Decreasing.
Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \).
The value of \( \int \cos^2 x \, dx \) will be:
Evaluate: \( \int_0^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \)
Prove that: \( \int_0^{\pi} \log(1 + \cos x) dx = -\pi \log_e 2 \)
Find the equations of the tangent and normal to the curve \( x^{2/3} + y^{2/3} = 2 \) at the point (1, 1).
Find the area of the region enclosed by the parabola \( y^2 = 4ax \) and its latus rectum.
The value of \( \int_1^{\sqrt{3}} \frac{dx}{1 + x^2} \) will be :
The value of \( \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be
(a) If \( x = a(\theta + \sin \theta) \), \( y = a(1 - \cos \theta) \), find \( \frac{dy}{dx} \):
(c) Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \):
(d) Prove that the function \( f(x) = |x| \) is not differentiable at \( x = 0 \):
(b) Find the differential coefficient of the function \( x^x \) with respect to \( x \):
If the normal of the curve \[ \frac{x^{2/3}}{a^{2/3}} + \frac{y^{2/3}}{a^{2/3}} = 1 \] makes an angle \( \theta \) with the x-axis, prove that the equation of the normal is \[ y \cos \theta - x \sin \theta = a \cos 2\theta. \]
Find the differential coefficient of \( \tan^{-1}\left( \frac{2x}{1-x^2} \right) \) \text{with respect to} \( x \).
Find the equation of normal at the point \( (1, 1) \) of the curve \( x^{2/3} + y^{2/3} = 2 \).