List of top Mathematics Questions on Calculus asked in PSEB XII

Prove that the function \( f : \mathbb{R} \to \mathbb{R} \), \( f(x) = \frac{5x + 3}{4} \), is one-one and onto.
Find \( \frac{dy}{dx} \), if \( y = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \)
Evaluate \( \int \frac{\sin^{-1}(x)}{1 + x^2} \, dx \)
Evaluate \( \int \frac{2x - 3}{x^2 + 1} \, dx \)
Using integration, find the area bounded by the circle \( x^2 + y^2 = 16 \) in the first quadrant.
Find the critical points of the function \( f(x) = 2x^3 - 3x^2 - 12x + 15 \):
Evaluate \( \int_{\pi/6}^{\pi/3} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \):
Evaluate \( \int_0^1 \frac{dx}{1 + x^2} \):
Find the order of the differential equation \( \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 3y = 0 \):
Evaluate \( \int \frac{x^2}{x^4+1} dx \).
Evaluate \( \int \frac{3x+2}{(x^2+1)(x-2)}dx \).
Solve: \( x \log x \frac{dy}{dx} + y = \frac{2}{x} \log x \).
Prove that the function, \( f: \mathbb{R} \to \mathbb{R}, f(x) = \frac{5x+3}{4} \) is one-one and onto.
Evaluate \( \int \frac{2x-3}{x^2+1} dx \).
Using integration, find the area bounded by the circle whose centre is at origin and radius is 4 units.
Order of differential equation \( \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 3y = 0 \) is:
\( \int_{\pi/6}^{\pi/3} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx \) is equal to:
\( \int_0^1 \frac{dx}{1+x^2} \) is equal to:
Relation \( R = \{(x, y) : x<y^2 \text{ where } x, y \in \mathbb{R}\} \) is: