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If \( \displaystyle \int_{\pi/6}^{\pi/4}\left(\cot\!\left(x-\dfrac{\pi}{3}\right)\cot\!\left(x+\dfrac{\pi}{3}\right)+1\right)dx=\alpha \log_{e}(\sqrt{3}-1) \), then \( 9\alpha^2 \) is equal to \( \_\_\_\_\_ \).

The value of the integral \( \displaystyle \int_{0}^{2}\dfrac{\sqrt{x(x^{2}+x+1)}}{\sqrt{x+1}\left(\sqrt{x^{4}+x^{2}+1}\right)}\,dx \) is equal to:

The value o the integral \( \int_{-\pi/4}^{\pi/4} \left( \frac{32\cos^4 x}{1+e^{\sin x}} \right) dx \) is:

Let \( f \) be a polynomial function such that \( f(x^{2}+1)=x^{4}+5x^{2}+2 \), for all \( x \in \mathbb{R} \). Then \( \displaystyle \int_{0}^{3} f(x) \, dx \) is equal to:

Let \( [\,\cdot\,] \) be the greatest integer function. If \( \alpha = \int_{0}^{64} \left(x^{1/3} - \left[x^{1/3}\right]\right) dx \), then \( \displaystyle \frac{1}{\pi} \int_{0}^{\alpha \pi} \left(\frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta}\right) d\theta \) is equal to

If \( f(x) \) satisfies the relation \( f(x)=e^x+\int_{0}^{1}(y+x e^x)\,f(y)\,dy \), then \( e+f(0) \) is equal to _______________.

The value of the integral \( \displaystyle \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}} \) is:

The value of \( \displaystyle \int_{0}^{20\pi}\left(\sin^{4}x+\cos^{4}x\right)dx \) is equal to:

Let [.] denote the greatest integer function. Then the value of \( \int_{0}^{3} (\frac{e^{x}+e^{-x}}{[x]!} \, dx \)) is:

The area of the region enclosed between the circles \( x^2 + y^2 = 4 \) and \( x^2 + (y-2)^2 = 4 \) is:

Let \( f \) be a twice differentiable function such that \( f(x)=\int_{0}^{x}\tan(t-x)\,dt-\int_{0}^{x}f(t)\tan t\,dt \), \( x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right) \). Then \( f'\!\left(\dfrac{\pi}{6}\right)+12f'\!\left(-\dfrac{\pi}{6}\right)+f\!\left(\dfrac{\pi}{6}\right) \) is equal to ________.

Let \( f : \mathbb{R} \to (0,\infty) \) be a twice differentiable function such that \( f(3)=18 \), \( f'(3)=0 \) and \( f''(3)=4 \). Then

The value of is -----------.

The area of the region \( R = \{(x, y) : xy \le 27,\, 1 \le y \le x^{2}\} \) is equal to:

The area of the region, inside the ellipse \( x^2 + 4y^2 = 4 \) and outside the region bounded by the curves \( y = |x| - 1 \) and \( y = 1 - |x| \), is:

The area of the region bounded by the curves \( x+3y^{2}=0 \) and \( x+4y^{2}=1 \) is equal to:

The area of the region \( \{(x,y) : 0 \le y \le 6 - x, \, y^{2} \ge 4x - 3, \, x \ge 0 \} \) 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:

The value of the integral \( \int_{0}^{\infty} \frac{\log_{e}(x)}{x^{2}+4} \, dx \) is:

Let \( f:(1,\infty)\to \mathbb{R} \) be a function defined as \( f(x)=\dfrac{x-1}{x+1} \). Let \( f^{\,i+1}(x)=f\!\left(f^{\,i}(x)\right),\ i=1,2,\ldots,25 \), where \( f^{1}(x)=f(x) \). If \( g(x)+f^{26}(x)=0,\ x\in(1,\infty) \), then the area of the region bounded by the curves \( y=g(x) \), \( 2y=2x-3 \), \( y=0 \) and \( x=4 \) is:

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 bounded by \( 16x^2 - 9y^2 = 144 \) and \( 8x - 2y = 24 \) is \( A \), then \( 3(A + 6\log_e(3)) \) is equal to ________.

\( 6 \displaystyle \int_{0}^{\pi} \left| (sin 3x + \sin 2x + \sin x) \right| dx \) is equal to __________.

If \( \alpha = \int_{0}^{2\sqrt{3}} \log_2 ({x^{2}+4}\,) dx + \int_{2}^{4} \sqrt{x^{2}-4}\, dx \), then \( \alpha^{2} \) is equal to ________.

The value of the integral \( \int_{\pi/6}^{\pi/3} (\frac{4-cosec^{2}x}{\cos^{4}x})\, dx \) is:

The integral \( \displaystyle \int_{0}^{1} \cot^{-1}(1 + x + x^{2}) \, dx \) is equal to:

Let the area of the region bounded by the curve \( y = \max\{\sin x , \cos x\} \), lines \( x = 0 \), \( x = \frac{3\pi}{2} \), and the x-axis be \( A \). Then, \( A + A^2 \) is equal to \( \_\_\_\_\_ \).

The number of elements in the set \( S = \{ x : x \in [0, 100] \text{ and } \int_{0}^{x} t^{2}\sin(x-t)\,dt = x^{2} \} \) is _________.

Let \( A_1 \) be the bounded area enclosed by the curves \( y = x^2 + 2 \), \( x + y = 8 \) and y-axis that lies in the first quadrant.
Let \( A_2 \) be the bounded area enclosed by the curves \( y = x^2 + 2 \), \( y^2 = x \), \( x = 2 \), and y-axis that lies in the first quadrant. Then \( A_1 - A_2 \) is equal to:

Let the line \( x = -1 \) divide the area of the region \( \{(x,y) : 1 + x^2 \le y \le 3 - x\} \) in the ratio \( m : n \), \( \gcd(m,n) = 1 \). Then \( m + n \) is equal to:

The area of the region \( \{(x,y): y\le \pi-|x|,\, |y|\le |\sin x|,\, y\ge 0\} \) is:

Let \( \int_{-2}^{2} \left(|\sin x| + [x\sin x]\right)\,dx = 2(3-\cos 2)+\beta \), where \( [\cdot] \) is the greatest integer function. Then \( \beta \sin\!\left(\dfrac{\beta}{2}\right) \) equals:

The value of \( \int_{-\pi/2}^{\pi/2}\frac{1}{[x]+4}\,dx \), where \( [ \, \cdot \, ] \) denotes the greatest integer function, is:

The value of the integral \( \displaystyle\int_{-1}^{1}\left(\dfrac{x^{3}+|x|+1}{x^{2}+2|x|+1}\right)dx \) is equal to:

If \( \alpha=1 \) and \( \beta=1+i\sqrt{2} \), where \( i=\sqrt{-1} \) are two roots of the equation \( x^3+ax^2+bx+c=0,\, a,b,c \in \mathbb{R} \), then is equal to:

Let \( e \) be the base of natural logarithm and let \(\alpha/\beta\) and \( g : \{1 \), \( e \), \( e^2 \), \( e^3 \} \to \{ 1 \), \( \tfrac{1}{2} \), \( \tfrac{1}{3} \), \( \tfrac{1}{4} \} \) be two bijective functions such that \( f \) is strictly decreasing and \( g \) is strictly increasing. If \( \phi(x)=\left[ f^{-1}\!\left\{\,g^{-1}\!\left( \tfrac{1}{2} \right)\!\right\}\right]^x \), then the area of the region \( R=\{(x,y): x^2 \le y \le \phi(x),\; 0 \le x \le 1\} \) is:

The value of is equal to:

Let a differentiable function \( f \) satisfy the equation. If \( \int_{0}^{36} f\!\left(\frac{t x}{36}\right) dt = 4 \alpha \, f(x) \). If \( y = f(x) \) is a standard parabola passing through the points \( (2, 1) \) and \( (-4, \beta) \), then \( \beta^{\alpha} \) is equal to _____.

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function such that \( f(x) + 3 f\!\left(\frac{\pi}{2} - x\right) = \sin x, \; x \in \mathbb{R} \). Let the maximum value of \( f \) on \( \mathbb{R} \) be \( \alpha \). If the area of the region bounded by the curves \( g(x) = x^2 \) and \( h(x) = \beta x^3, \; \beta > 0 \), is \( \alpha^2 \), then \( 30\beta^3 \) is equal to ________.

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JEE Main 2026 21st Jan Morning Shift

The value of is equal to: