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

Integrate: \( \int \left(\frac{2 + \sin 2x}{1 + \cos 2x}\right) e^x dx \)

Prove that \(\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\tan x}} = \frac{\pi}{12}\).

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.

Prove that \( \int_{0}^{\pi/2} \sqrt{\frac{1+\cos 4x}{2}} \, dx = 1 \).

Evaluate: \(\int \sqrt{3 - 2x - x^2} dx\).

Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x>0 \).

If \(y=(\tan^{-1} x)^2\), show that \((x^2+1)^2 \frac{d^2y}{dx^2} + 2x(x^2+1)\frac{dy}{dx} = 2\).

Prove that \(\int_0^{\pi/2} \log(\cos x) \, dx = -\frac{\pi}{2} \log 2\).

Prove that \(\int_0^\pi \sqrt{\frac{1+\cos 2x}{2}} \, dx = 2\).

At \( t = 2 \), the slope of the vector function \( \vec{f}(t) = 2\hat{i} + 3\hat{j} + 5t^2\hat{k} \) is

If function \( f \) is defined as \[ f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \] then prove that \( f \) is continuous.

The vector function is given by \(\vec{f}(t) = t\hat{i} + t^2\hat{j} + 5\hat{k}\), then at point \(t = 1\) the slope is

If \(\int x \log x \, dx = \frac{x^2}{2} f(x) - \frac{x^2}{4} + c\), then \( f(x) \) is

Prove that \(\int_{0}^{\pi/4} \log(1 + \tan x) \, dx = \frac{\pi}{8} \log 2\).

Show that the given function \( f, f(x) = x^3 - 3x^2 + 4x, x \in \mathbb{R} \) is an increasing function in \( \mathbb{R} \).

Find the value of the determinant \( \begin{vmatrix} x+y+2z & x & y \\ z & y+z+2x & y \\ z & x & z+x+2y \end{vmatrix} \).

Evaluate: \( \int \frac{dx}{(x+1)(x+2)} \).

Find the value of the determinant \( \begin{vmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ca & cb & c^2+1 \end{vmatrix} \).

Prove that \(\tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1}x\), where \(-\frac{1}{\sqrt{2}} \leq x \leq 1\).

Test whether the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \[ f(x) = \begin{cases} x+5, & \text{if } x \le 1 \\ x-5, & \text{if } x > 1 \end{cases} \] is continuous at \(x=1\).

If \( I = \int_{0}^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \), find the value of I.

If \( I = \int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}} \), then prove that \( I = \frac{\pi}{12} \).

A car is started to move 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 \(x = t^2\left(3-\frac{t}{2}\right)\). Find the time required by the car to reach at the point Q and also find the distance between P and Q.

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.