List of top Mathematics Questions on Calculus asked in JEE Main

If domain of $f(x) = \sqrt{\log_{0.6} \frac{2x - 5}{x^2 - 4}}$ is $(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)$ then the value of $(a + b + c + d + e)$ is :
Consider $f(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases}$
Number of critical points of $f(x)$ in the interval $(-2\pi, 2\pi)$ is:
The area of the region \[ A = \{(x,y) : 4x^2 + y^2 \le 8 \;\text{and}\; y^2 \le 4x\} \] is
The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{12(3+[x])\,dx}{3+[\sin x]+[\cos x]} \] (where \([\,]\) denotes the greatest integer function) is:
If \[ f(x)=1-2x+\int_{0}^{x} e^{x-t} f(t)\,dt \] and \[ g(x)=\int_{0}^{x} (f(t)+2)^{11}(t+12)^{17}(t-4)^4\,dt, \] If local minima and local maxima of \(g(x)\) occur at \(x=p\) and \(x=q\) respectively, find \(|p|+q\).
Let the line \( x = -1 \) divide the area of the region \[ \{(x,y): 1 + x^2 \le y \le 3 - x\} \] in the ratio \( m:n \), where \( \gcd(m,n)=1 \). Then \( m+n \) is equal to
Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3), x \in \mathbb{R}$. Then the value of $f'(5)$ is :
Let \[ f(t)=\int_{0}^{t} e^{x^2}\Big((1+2x^2)\sin x+x\cos x\Big)\,dx. \] Then the value of \(f(\pi)-f\!\left(\frac{\pi}{2}\right)\) is equal to:
Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x=0$, $x=1$, $y^2=x$ and $y=|\alpha x-5|-|1-\alpha x|+\alpha^2$. Then $(f(0)+f(1))$ is equal to
A rectangle is formed by the lines \( x = 0 \), \( y = 0 \), \( x = 3 \) and \( y = 4 \). Let the line \( L \) be perpendicular to \( 3x + y + 6 = 0 \) and divide the area of the rectangle into two equal parts. Then the distance of the point \( \left(\frac{1}{2}, -5\right) \) from the line \( L \) is equal to :
Let \(\int \frac{1 - 5 \cos^2 x}{\sin^5 x \cos^2 x} dx = f(x) + c\) then find \(f\left(\frac{\pi}{4}\right) - f\left(\frac{\pi}{6}\right)\) :
The value of \(S = \sum_{r=1}^{20} \sqrt{\pi \int_{0}^{r} x |\sin \pi x| dx}\) is :
If \(f(x^2 + 1) = x^4 + 5x^2 + 1\), then find \(\int_{0}^{3} f(x) dx\) :
The value of the integral \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4}\,dx, \] where $[\cdot]$ denotes the greatest integer function, is
Let \(x \frac{dy}{dx} - \sin 2y = x^3(2 - x^3) \cos^2 y ; y(2) = 0\), then find \(\tan(y(1))\) :
Given \[ f(x)=\int \frac{dx}{x^{2/3}+2\sqrt{x}} \quad \text{and} \quad f(0)=-26+24\ln 2. \] If \(f(1)=A+B\ln 3\), then find \((A+B)\).
Area bounded by \(\{(x, y) : xy \le 8 ; y \le x^2, y \ge 1\}\) is :
Consider two parabolas \(P_1,\ P_2\) and a line \(L\):
\[ P_1:\ y=4x^2,\qquad P_2:\ y=x^2+27,\qquad L:\ y=\alpha x \] If the area bounded by \(P_1\) and \(P_2\) is six times the area bounded by \(P_1\) and \(L\), find \(\alpha\).
Value of : \(\sum_{k=1}^{\infty} \frac{(-1)^k \cdot k(k+1)}{k!}\)
The area of the region \[ R=\{(x,y): xy\le 8,\; 1\le y\le x^2,\; x\ge 0\} \] is:

If the area of the region \[ \{(x, y) : 1 - 2x \le y \le 4 - x^2,\ x \ge 0,\ y \ge 0\} \] is \[ \frac{\alpha}{\beta}, \] \(\alpha, \beta \in \mathbb{N}\), \(\gcd(\alpha, \beta) = 1\), then the value of \[ (\alpha + \beta) \] is : 

If \( \int_0^1 4 \cot^{-1}(1 - 2x + 4x^2) dx = a \tan^{-1}(2) - b \ln(5) \), where \( a, b \in \mathbb{N} \), then \( (2a + b) \) is equal to _________.
If $f(a)$ is the area bounded in the first quadrant by $x=0$, $x=1$, $y=x^2$ and $y=|ax-5|-|1-ax|+ax^2$, then find $f(0)+f(1)$.
The value of \(\lim_{x \to 0} \frac{\ln(\sec(ex) \cdot \sec(e^2x) \dots \sec(e^{10}x))}{e^2 - e^{2\cos x}}\) is :
If $f(a)$ is the area bounded in the first quadrant by $x=0$, $x=1$, $y=x^2$ and $y=|ax-5|-|1-ax|+ax^2$, then find $f(0)+f(1)$.