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Let \( g(x) = \displaystyle\int\limits_{0}^{x} f(t)\,dt \), where \( f \) is continuous function in \( [0,3] \) such that \( \dfrac{1}{3} \le f(t) \le 1 \) for all \( t \in [0,1] \) and \( 0 \le f(t) \le \dfrac{1}{2} \) for all \( t \in [1,3] \). The largest possible interval in which \( g(3) \) lies is:

The value of the integral \( \displaystyle \int_{0}^{1}\frac{\sqrt{x}\,dx}{(1+x)(1+3x)(3+x)} \) is:

Let \( I_{n}=\displaystyle\int_{1}^{e}x^{19}(\log |x|)^{n}\,dx \), where \( n\in \mathbb{N} \). If \( (20)\,I_{10}= \alpha I_{9}+\beta I_{8} \) for natural numbers \( \alpha \) and \( \beta \), then \( \alpha-\beta \) equals to ________.

Let \( f:[0,\infty)\rightarrow [0,\infty) \) be defined as \( f(x)=\displaystyle\int\limits_{0}^{x}[ y]r \,dy \), where \( [ x] \) is the greatest integer less than or equal to \( x \). Which of the following is true?

If \( \displaystyle \int_{0}^{{\pi}{}} (\sin^{3}x )\, e^{-\sin^{2}x}\,dx = \alpha - \dfrac{\beta}{e}\displaystyle \int_{0}^{1} \sqrt{t}\,e^{t}\,dt \), then \( \alpha + \beta \) is equal to ____________.

For real numbers \( \alpha, \beta, \gamma \) and \( \delta \), if
\( \int \dfrac{(x^{2}-1) + \tan^{-1}\!\left(\dfrac{x^{2}+1}{x}\right)}{(x^{4}+3x^{2}+1)\,\tan^{-1}\!\left(\dfrac{x^{2}+1}{x}\right)} \, dx = \alpha \log_{e}\!\left( \tan^{-1}\!\left(\dfrac{x^{2}+1}{x}\right) \right) + \beta \tan^{-1}\!\left( \dfrac{\gamma (x^{2}-1)}{x} \right) + \delta \tan^{-1}\!\left( \dfrac{x^{2}+1}{x} \right) + C \) where \( C \) is an arbitrary constant, then the value of \( 10(\alpha+\beta\gamma+\delta) \) is equal to ________.

$\text{The value of } \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{1 + \sin^2 x}{1 + \pi^{\sin x}}\right) dx \text{ is:}$

Let \( p(x)=x^{2}+bx+c \) be a quadratic polynomial with real coefficients such that \( \int\limits_{0}^{1} p(x)\,dx = 1 \) and \( p(x) \) leaves remainder \( 5 \) when it is divided by \( (x-2) \). Then the value of \( 9(b+c) \) is equal to:

The area, enclosed by the curves \( y = \sin x + \cos x \) and \( y = |\cos x - \sin x| \) and the lines \( x = 0 \), \( x = \frac{\pi}{2} \), is:

If a curve \( y = f(x) \) passes through the point \( (1, 2) \) and satisfies \( x \, \frac{dy}{dx} + y = b x^4 \), then for what value of \( b \), \( \int\limits_{1}^{2} f(x) \, dx = \frac{62}{5} \)?

Let \( [t] \) denote the greatest integer \( \leq t \). Then the value of \( 8 \cdot \int_{-1/2}^{1} \big([2x] + |x|\big)\,dx \) is __________ .

Let \( F:(3,5)\to \mathbb{R} \) be a twice differentiable function on \( (3,5) \) such that \( F(x)=e^{-x}\displaystyle \int_{3}^{x}\big(3t^{2}+2t+4F'(t)\big)\,dt \). If \( F'(4)=\dfrac{\alpha e^{\beta}-224}{\left(e^{\beta}-4\right)^{2}} \), then \( \alpha+\beta \) is equal to _________.

Consider the integral \( I=\displaystyle\int\limits_{0}^{10}\dfrac{[x] \, e^{[x]}}{e^{x-1}}\,dx \), where \( [x] \) denotes the greatest integer less than or equal to \( x \). Then the value of \( I \) is equal to:

The value of \( \displaystyle\lim_{n \to \infty} \dfrac{1}{n} \sum\limits_{j=1}^{n} \dfrac{(2{\,j-1)} + 8n}{(2{\,j-1)} + 4n} \) is equal to:

The area of the region \( S = \{(x, y) : 3x^{2} \le 4y \le 6x + 24\} \) is ________.

The value of the integral \( \displaystyle \int\limits_{-1}^{1}\log\!\left(x+\sqrt{x^{2}+1}\right)\,dx \) is:

The value of the integral \( \displaystyle \int\limits_{-1}^{1}\log_{e}\!\left(\sqrt{1-x}+\sqrt{1+x}\right)\,dx \) is equal to:

Let \( f(x) \) be a differentiable function defined on \( [0,2] \) such that \( f'(x)=f'(2-x) \) for all \( x \in (0,2) \), \( f(0)=1 \) and \( f(2)=e^{2} \). Then the value of \( \int\limits_{0}^{2} f(x)\,dx \) is:

The area (in sq. units) of the region, given by the set \( \{(x,y) \in R \times R \mid x \ge 0, 2x^2 \le y \le 4 - 2x\} \) is:

If the line \( y = mx \) bisects the area enclosed by the lines \( x = 0 \), \( y = 0 \), \( x = \tfrac{3}{2} \) and the curve \( y = 1 + 4x - x^2 \), then \( 12m \) is equal to ________.

If $[x]$ is the greatest integer $\le x$, then $\pi^2 \int_{0}^{2} \left(\sin \frac{\pi x}{2}\right) (x - [x])^{[x]} dx$ is equal to:

If the value of the integral \( \displaystyle \int_{0}^{5} \frac{x + \lfloor x \rfloor}{e^{\,x - \lfloor x \rfloor}} \, dx = \alpha e^{-1} + \beta \), where \( \alpha, \beta \in \mathbb{R} \), \( 5\alpha + 6\beta = 0 \), and \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \); then the value of \( (\alpha + \beta)^{2} \) is equal to:

The area bounded by the curve \( 4y^2 = x^2 (4 - x)(x - 2) \) is equal to:

The area of the region bounded by the parabola \( (y - 2)^2 = (x - 1) \), the tangent to it at the point whose ordinate is \( 3 \) and the \( x \)-axis is :

The area (in sq. units) of the region bounded by the curves \( x^2 + 2y - 1 = 0 \), \( y^2 + 4x - 4 = 0 \) and \( y^2 - 4x - 4 = 0 \), in the upper half plane is _______.

Let \( a \) be a positive real number such that \( \displaystyle\int\limits_{0}^{a} e^{\,x-[x]}\,dx = 10e - 9 \) Where \( [x] \) is the greatest integer less than or equal to \( x \). Then \( a \) is equal to:

The value of the definite integral \( \displaystyle \int\limits_{-\pi/4}^{\pi/4} \frac{{d x}}{(1 + e^{x \cos x})(\sin^4 x + \cos^4 x)} \, dx \) is equal to:

Let \( a \) and \( b \) respectively be the points of local maximum and local minimum of the function \( f(x)=2x^{3}-3x^{2}-12x \).
If \( A \) is the total area of the region bounded by \( y=f(x) \), the \( x \)-axis and the lines \( x=a \) and \( x=b \), then \( 4A \) is equal to __________.

The value of \( \displaystyle \lim_{n \to \infty} \dfrac{1}{n} \displaystyle \sum_{r=0}^{2n-1} \dfrac{n^{2}}{n^{2}+4 r^{2}} \) is:

If \([\,\cdot\,]\) represents the greatest integer function, then the value of \( \left|\displaystyle \int_{0}^{\sqrt{\pi/2}}\!\big[[x^{2}] - \cos x\big]\,dx \right| \) is __________.

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = e^{-x} \sin x \). If \( F : [0,1] \to \mathbb{R} \) is a differentiable function such that \( F(x) = \displaystyle\int\limits_{0}^{x} f(t)\, dt \), the value of \( \displaystyle \int\limits_{0}^{1} \left(F^{\prime}(x) + f(x)\right)e^{x} dx \) lies in the interval.

Let the domain of the function \( f(x)=\log_4\!\Big(\log_5\!\big(\log_3(18x-x^2-77)\big)\Big) \) be \( (a,b) \). Then the value of the integral \( \displaystyle \int_a^b \frac{\sin^3 x}{(\sin^3 x+\sin^3(a+b-x))}\,dx \) is equal to \( \underline{\quad} \).

The function \( f(x) \), that satisfies the condition \( f(x) = x + \int_{0}^{\pi/2} \sin x \cdot \cos y \; f(y)\, dy \), is:

In the integral \( \displaystyle \int_{0}^{10} \frac{[\sin 2\pi x]}{e^{\,x-[ x ]}} \, dx = \alpha e^{-1} + \beta e^{-1/2} + \gamma \), where \( \alpha, \beta, \gamma \) are integer and \( [x] \) denotes the greatest integer less than or equal to \( x \), then the value of \( \alpha + \beta + \gamma \) is equal to:

If the area of the bounded region \( R = \left\{ (x,y) : \max \{0, \log_e x \}\le y \le 2^x, \ \dfrac{1}{2} \le x \le 2 \right\} \) is, \( \alpha (\log_e 2)^{-1} + \beta \log_e 2 + \gamma \), then the value of \( (\alpha + \beta - 2\gamma)^2 \) is equal to:

Which of the following statements is incorrect for the function \( g(\alpha) \) for \( \alpha \in \mathbb{R} \) such that
\( g(\alpha) = \int_{\pi/6}^{\pi/3} \frac{\sin^{\alpha} x}{\cos^{\alpha} x + \sin^{\alpha} x} \, dx \)

If \( U_n=\left(1+\frac{1}{n^2}\right)^1\left(1+\frac{2^2}{n^2}\right)^2\cdots\left(1+\frac{n^2}{n^2}\right)^n \), then \( \displaystyle \lim_{n\to \infty}(U_n)^{-\frac{4}{n^2}} \) is equal to:

Let \( f(x) \) and \( g(x) \) be two functions satisfying \( f(x^2) + g(4 - x) = 4x^3 \) and \( g(4 - x) + g(x) = 0 \), then the value of \( \displaystyle \int_{-4}^{4} f(x^2) \, dx \) is ____________.

Let \( f:[-3,1]\to R \) be given as If the area bounded by \( y=f(x) \) and x-axis \( A \), then the value of \( 6A \) is equal to _______.

The value of the integral, \( \int_{1}^{3} [x^{2}-2x-2] \, dx \), when \([x]\) denotes the greatest integer less than or equal to \( x \), is:

The value of $\int_{-1/\sqrt{2}}^{1/\sqrt{2}} \left( \left(\frac{x+1}{x-1}\right)^2 + \left(\frac{x-1}{x+1}\right)^2 - 2 \right)^{1/2} dx$ is:

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

The area of the region : \( R=\{(x.y):5x^2 \le y \le 2x^2+9\} \) is:

\( \displaystyle \int_{6}^{16} \dfrac{\log_{e} x^{2}}{\log_{e} x^{2} + \log_{e}(x^{2} - 44x + 484)} \, dx \) is equal to:

Let \( f \) be a non-negative function in \( [0,1] \) and twice differentiable in \( (0,1) \). If \( \int_{0}^{x} \sqrt{1-(f'(t))^{2}} \, dt = \int_{0}^{x} f(t)\, dt, \; 0 \le x \le 1 \) and \( f(0)=0 \), then \( \displaystyle \lim_{x \to 0} \dfrac{1}{x^{2}} \int_{0}^{x} f(t)\, dt \):

The area of the region bounded by \( y - x = 2 \) and \( x^{2} = y \) is equal to:

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JEE Main 2021 25th July Morning Shift

Let \( f:[0,\infty)\rightarrow [0,\infty) \) be defined as \( f(x)=\displaystyle\int\limits_{0}^{x}[ y]r \,dy \), where \( [ x] \) is the greatest integer less than or equal to \( x \). Which of the following is true?