List of top Mathematics & Statistics Questions on Calculus

Write the condition for the function \( f(x) \), to be strictly increasing, for all \( x \in \mathbb{R} \).
Show that: \( \int_0^\pi \log (1 + \tan x) \, dx = \frac{\pi}{8} \log 2 \)
Find the area of region between parabolas \( y^2 = 4ax \) and \( x^2 = 4ay \).
A wire of length 36 meters is bent to form a rectangle. Find its dimensions if the area of the rectangle is maximum.
If \( y = e^{m \tan^{-1} x} \), then show that \( (1 + x^2) \left( \frac{dy}{dx} \right)^2 + (2x - 2m) \frac{dy}{dx} = 0 \).
Evaluate: \( \int \sec^3 x \, dx \)
Find the equation of tangent to the curve \( y = 2x^3 + x^2 + 2 \) at \( \left( \frac{1}{2}, 2 \right) \).
The maximum value of the function \( f(x) = \frac{\log x}{x} \) is
If \( f(x) = x^5 + 2x - 3 \), then \( (f^{-1})'(-3) = \)
If \( x = f(t) \) and \( y = g(t) \) are differentiable functions of t so that y is a differentiable function of x and \( \frac{dx}{dt} \neq 0 \), then prove that \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \). Hence find \( \frac{dy}{dx} \) if \( x = \sin t \) and \( y = \cos t \).
If u and v are differentiable functions of x, then prove that \( \int u \, dv = uv - \int v \, du \). Hence evaluate \( \int \log x \, dx \).
Evaluate: \( \int \frac{dx}{2 + \cos x - \sin x} \)
Show that function \( f(x) = \tan x \) is increasing in \( (0, \frac{\pi}{2}) \).
If \( \int \frac{dx}{4x - 1} = A \log |2x + 1| + c \), then \( A = \)
If \( f'(x) = x^{-1} \), then find \( f(x) \)