List of top Mathematics Questions

\( \int_0^{\pi} x \left( \sin^2(\sin x) + \cos^2(\cos x) \right) dx = \)
Evaluate \( \int_{-\pi}^{\pi} \frac{\cos^{2022}x}{1 + (2022)^x} \, dx \).
If $f(x) = \begin{cases} \dfrac{x^2 \log(\cos x)}{\log(1+x)}, & x \neq 0 \\ 0, & x = 0 \end{cases}$, then at $x=0$, $f(x)$ is
If \( I_n = \int \tan^n x \, dx \), and \( I_0 + I_1 + 2I_2 + 2I_3 + 2I_4 + I_5 + I_6 = \sum_{k=1}^n \frac{\tan^k x}{k} \), then \( n = \)
\( \lim_{x \to a} \frac{x^3+a^3}{x+a} = 7 \implies a = \)
Given \[ \frac{d}{dx} \left( \frac{(x+1)^2 \sqrt{x-1}}{(x+4)^3 e^x} \right) = f(x) \left[ \frac{2}{x+1} + \frac{1}{2(x-1)} - \frac{3}{x+4} - 1 \right] \] Then find \( f(5) \).
Two particles P and Q located at the points \( P(t, t^3 - 16t - 3) \), \( Q(t+1, t^3 - 6t - 6) \) are moving in a plane. The minimum distance between the points in their motion is:
Which of the following functions is differentiable at \( x = 0 \)?
The equation of the line which is a normal to the curve \(3x^2-xy+y^2=3\) at \((1,1)\) is
The minimum value of \( f(x) = x + \frac{4}{x + 2} \) is
\(\tanh(\log x) = \) (Assuming \(\log x\) is natural logarithm \(\ln x\))
For the differential equation \( \frac{d^3 y}{dx^3} = 0 \), \( y = ax^2 + bx + c \) is:
\( \int_{-\pi/4}^{\pi/4} \cos^8 x \ dx = \)
The minimum distance of a point on the curve \( y = x^2 - 4 \) from the origin is
At any point \( (x, y) \) on a curve, if the length of the subnormal is \( (x - 1) \) and the curve passes through \( (1, 2) \), then the curve is a conic. A vertex of the curve is:
Evaluate: \( \int \frac{1 + \tan x \tan(x + \alpha)}{\tan x \tan(x + \alpha)} dx \):
If $x \neq 0$ and $f(x)$ satisfies $8f(x) + 6f\left(\dfrac{1}{x}\right) = x + 5$, then $\dfrac{d}{dx} \left(x^2 f(x)\right)$ at $x = 1$ is
Evaluate: \[ \int_{0}^{\pi/4} e^{\tan^2 \theta} \sin^2 \theta \tan \theta \, d\theta = \]
Assertion (A): $$ \frac{d}{dx} \left( \frac{x^2 \sin x}{\log x} \right) = \frac{x^2 \cos x \cdot \log x - x^2 \sin x \cdot \frac{1}{x}}{(\log x)^2} + \frac{2x \sin x}{\log x} \Rightarrow \text{(matches the structure of product + quotient rule)} $$ Reason (R): $$ \frac{d}{dx} \left( \frac{uv}{w} \right) = \frac{uv'}{w} + \frac{u'v}{w} - \frac{uvw'}{w^2} \Rightarrow \text{(correct general derivative formula)} $$
If \( f(x) = \int_{x^2}^{\cos^2 x} (2x \tan^2 x - 2x - 6 \tan x) dx \), and \( f(0) = \pi \), then \( f(x) = \):
\( \lim_{x \to 0} x^3 (\sqrt{x^2+\sqrt{x^4+1}} - \sqrt{2}x) = \)
Equation of tangent to the curve \( y = x + \frac{4}{x^2} \) which is parallel to x-axis is
The normal to the curve y = f(x) at the points (3, 4) makes an angle \( \frac{3\pi}{4} \) with positive x-axis then \(f'(3) = \)
\( \int \frac{dx}{(1+\sqrt{x})^{2022}} = \)
The maximum value of \( f(x) = \frac{x}{1 + 4x + x^2} \) is