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Let for some function \( y=f(x) \), \( \int_{0}^{x} t f(t)\,dt = x^{2} f(x), \, x>0 \) and \( f(2)=3 \). Then \( f(6) \) is equal to:

Let the area of region \( \{(x, y) : 2y \le x^2 + 3,\; y + |x| \le 3,\; y \ge0| x - 1|\} \) be \( A \). Then \( 6A \) is equal to:

The value of $\int\limits_{-1}^{1} \frac{(1 + \sqrt{|x| - x})e^x + (\sqrt{|x| - x})e^{-x}}{e^x + e^{-x}} dx$ is equal to:

If \( \int_{-\pi/2}^{\pi/2} \frac{96 x^{2}\cos^{2}x}{(1+e^{x})} \, dx = \pi(\alpha \pi^{2} + \beta), \\ \alpha, \beta \in \mathbb{Z}, \) then \( (\alpha + \beta)^{2} \) equals:

A line passing through the point \( A(-2,0) \), touches the parabola \( P: y^{2}=x-2 \) at the point \( B \) in the first quadrant. The area, of the region bounded by the line \( AB \), parabola \( P \) and the x-axis is:

The area of the region bounded by the curves \( x(1+y^2)=1 \) and \( y^2=2x \) is:

The value of $\int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{\left((\log_e x)^2 + 1\right)^{-1}}}{e^{\left((\log_e x)^2 + 1\right)^{-1}} + e^{\left((6 - \log_e x)^2 + 1\right)^{-1}}} \right) dx$ is :

The area (in sq. units) of the region \( \{(x, y) : 0 \le y \le 2|x| + 1, \ 0 \le y \le x^2 + 1, \ |x| \le 3\} \) is:

Let \( x = x(y) \) be the solution of the differential equation \( y = \left(x - y \frac{dx}{dy}\right) \sin \left(\frac{x}{y}\right) \), \( y > 0 \) and \( x(1) = \frac{\pi}{2} \). Then \( \cos x(2) \) is equal to:

If the area of the region \( \{(x,y): |4-x^{2}|\le y \le x^{2},\, y\le 4,\, x\ge 0\} \) is \( \left(\frac{80\sqrt{2}}{\alpha}-\beta\right) \), \( \alpha, \beta \in \mathbb{N} \), then \( \alpha+\beta \) is equal to __________.

The area of the region, inside the circle \( (x - 2\sqrt{3})^{2} + y^{2} = 12 \) and outside the parabola \( y^{2} = 2\sqrt{3}\,x \) is:

Let $[\cdot]$ denote the greatest integer function. If $\int_{0}^{e^3} \left[\frac{1}{e^{x-1}}\right] dx = \alpha - \log_e 2$, then $\alpha^3$ is equal to _______.

Let the area of the bounded region \( \{(x, y) : 0 \leq 9x \leq y^2, \, y \geq 3x - 6\} \) be \( A \). Then \( 6A \) is equal to ______.

If \(\displaystyle \lim_{t \to 0} \left[ \int_{0}^{1} (3x+5)^{t} \, dx \right]^\dfrac{1}{t} = \frac{\alpha}{5e}\left(\frac{8}{5}\right)^\dfrac{2}{3} \), then \( \alpha \) is equal to ________.

Let \( f(x) \) be a positive function and \( I_1 = \int_{-1/2}^{1} 2x\, f\!\big(2x(1-2x)\big)\,dx \) and \( I_2 = \int_{-1}^{2} f\!\big(x(1-x)\big)\,dx \). Then the value of \( \dfrac{I_2}{I_1} \) is equal to ______.

If the area of the region \( \{(x,y): -1 \le x \le 1,\, 0 \le y \le a + e^{|x|} - e^{-x},\, a > 0\} \) is \( \frac{e^2 + 8e + 1}{e} \), then the value of \( a \) is:

Let \( f:\mathbb{R}\to\mathbb{R} \) be a twice differentiable function such that \( f(2)=1 \). If \( F(x)=x f(x) \) for all \( x\in\mathbb{R} \), \(\displaystyle \int_{0}^{2} xF'(x)\,dx = 6 \) and \(\displaystyle \int_{0}^{2} x^{2}F''(x)\,dx = 40 \), then \(\displaystyle F'(2)+\int_{0}^{2} F(x)\,dx \) is equal to:

Let the domain of the function \( f(x) = \log_{2} \log_{4} \log_{6} (3 + 4x - x^2) \) be \( (a, b) \). If \( \int_{0}^{b-a} [x^2] \, dx = p - \sqrt{q} - \sqrt{r}, \, p, q, r \in \mathbb{Z}, \, \gcd(p,q,r) = 1 \), where \( [\cdot] \) is the greatest integer function, then \( p + q + r \) is equal to:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f(x+y)=f(x)f(y) \) for all \( x,y \in \mathbb{R} \). If \( f'(0)=4a \) and \( f \) satisfies \( f''(x)-3a f'(x)-f(x)=0, \, a>0 \), then the area of the region \( R=\{(x,y)\mid 0\le y\le f(ax),\,0\le x\le 2\} \) is:

Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \); \(\beta,\gamma\) are integers, then the value of \( |\beta - \gamma| \) equals.

The integral \( \int_{-1}^{3/2} \big(| \pi^{2} x \sin(\pi x) \big|) \, dx \) is equal to:

Let \( f:[0,\infty) \to \mathbb{R} \) be a differentiable function such that \( f(x)=1-2x+\int_{0}^{x} e^{x-t}f(t)\,dt \) for all \( x \in [0,\infty) \). Then the area of the region bounded by \( y=f(x) \) and the coordinate axes is:

Let \( (a,b) \) be the point of intersection of the curve \( x^{2} = 2y \) and the straight line \( y - 2x - 6 = 0 \) in the second quadrant. Then the integral \( I = \int_{a}^{b} \dfrac{9x^{2}}{1 + 5^{x}} \, dx \) is equal to:

The area of the region \( \{(x,y): |x - y| \le y \le 4\sqrt{x} \} \) is:

Let \( f : (0,\infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \ne 0 \), \( \displaystyle \int_{0}^{1} f(\lambda x)\, d\lambda = a f(x) \) and \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \) then \( 16 - f'\!\left(\frac{1}{16}\right) \) is equal to ______.

If \(\displaystyle \int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_{e}\left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| + C \), where \( C \) is the constant of integration, then \( \alpha + 2\beta \) is equal to ____________.

The integral \( 80 \int_{0}^{\pi/4} (\frac{\sin \theta+\cos \theta}{9+16\sin 2\theta})\, d\theta \)is equal to:

Let \( f(x)+2f\!\left(\dfrac{1}{x}\right)=x^2+5 \) and \( 2g(x)-3g\!\left(\dfrac{1}{2}\right)=x,\; x>0 \). If \( \alpha=\displaystyle \int_{1}^{2} f(x)\,dx \) and \( \beta=\displaystyle \int_{1}^{2} g(x)\,dx \), then the value of \( 9\alpha+\beta \) is:

If the area of the region bounded by the curves \( y = 4 - \frac{x^2}{4} \) and \( y = \frac{x - 4}{2} \) is equal to \( \alpha \), then \( 6\alpha \) equals:

The area of the region enclosed by the curves \( y=e^{x} \), \( y=|e^{x}-1| \) and y-axis is:

If the area of the larger portion bounded between the curves \( x^{2} + y^{2} = 25 \) and \( y = |x - 1| \) is \( \dfrac{1}{4}(b\pi + c), \; b, c \in \mathbb{N} \), then \( b + c \) is equal to ________.

The integral $\int\limits_{0}^{\pi} \frac{(x + 3) \sin x}{1 + 3\cos^2 x} dx$ is equal to:

If \( I(m, n) = \int_{0}^{1} x^{m-1} (1 - x)^{n-1} \, dx, \ m, n > 0 \), then \( I(9, 14) + I(10, 13) \) is:

The area of the region bounded by the curve \( y = \max\{ |x|,x |x - 2| \} \), the \( x \)-axis and the lines \( x = -2 \) and \( x = 4 \) is equal to _________.

If \( 24\displaystyle\int_{0}^\frac{\pi}{4}\!\left(\sin\left|4x-\dfrac{\pi}{12}\right|+[ 2\sin x]\right)dx=2\pi+\alpha \), where \( [.]\) denotes the greatest integer function, then \( \alpha \) is equal to _________.

Let for \( f(x) = 7 \tan^{8} x + 7 \tan^{6} x - 3 \tan^{4} x - 3 \tan^{2} x \), \( I_{1} = \int_{0}^{\pi/4} f(x) \, dx \) and \( I_{2} = \int_{0}^{\pi/4} x \, f(x) \, dx \). Then \( 7 I_{1} + 12 I_{2} \) is equal to:

Let the function,

be differentiable for all \( x\in \mathbb{R} \), where \( a>1, b\in \mathbb{R} \). If the area of the region enclosed by \( y=f(x) \) and the line \( y=-20 \) is \( \alpha+\beta\sqrt{3} \), \( \alpha,\beta\in \mathbb{Z} \), then the value of \( \alpha+\beta \) is ______.

The area of the region \( \{(x,y): x^{2}+4x+2 \le y \le |x+2|\} \) is equal to:

If \(\displaystyle I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x} \, dx \) then \(\displaystyle \int_{0}^{2I} \frac{x \sin x \cos x}{\sin^{4} x + \cos^{4} x} \, dx \) equals:

If the area of the region \( \{(x,y) : |x - 5| \le y \le 4\sqrt{x} \} \) is \( A \), then \( 3A \) is equal to ______.

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JEE Main 2025 23rd Jan Evening Shift

Let \( x = x(y) \) be the solution of the differential equation \( y = \left(x - y \frac{dx}{dy}\right) \sin \left(\frac{x}{y}\right) \), \( y > 0 \) and \( x(1) = \frac{\pi}{2} \). Then \( \cos x(2) \) is equal to: