Integral Calculus — JEE Main PYQs

The value of $\int_0^{20\pi} (\sin^4 x + \cos^4 x)\,dx$ is equal to :

Let $f(x) = \int\left(\frac{16x + 24}{x^2 + 2x - 15}\right)dx$. If $f(4) = 14\log_e(3)$ and $f(7) = \log_e(2^{\alpha} \cdot 3^{\beta})$, $\alpha, \beta \in \mathbb{N}$, then $\alpha + \beta$ is equal to :

If the area of the region bounded by $16x^2 - 9y^2 = 144$ and $8x - 3y = 24$ is A, then $3(A + 6\log_e(3))$ is equal to __________.

The area of the region $\{(x, y) : y \leq \pi - |x|,\ y \leq |x \sin x|,\ y \geq 0\}$ 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 $\beta \sin\left(\frac{\beta}{2}\right)$ equals:

The area of the region bounded by the curves $x + 3y^2 = 0$ and $x + 4y^2 = 1$ is equal to:

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

Let $f$ be a twice differentiable function such that

$$f(x) = \int_{0}^{x} \tan(t - x)\,dt - \int_{0}^{x} f(t)\tan t\,dt, \quad x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right).$$

Then $f''\left(\dfrac{\pi}{6}\right) + 12\,f'\left(-\dfrac{\pi}{6}\right) + f\left(\dfrac{\pi}{6}\right)$ is equal to __________

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

The value of the integral $\displaystyle\int_0^{\infty} \frac{\log_e(x)}{x^2 + 4} \, dx$ is:

The value of the integral $\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left( \frac{4 - \operatorname{cosec}^2 x}{\cos^4 x} \right) dx$ is:

Let $(2^{1-a} + 2^{1+a})$, $f(a)$, $(3^a + 3^{-a})$ be in A.P. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\displaystyle\int_{\log_e(\alpha - 1)}^{\log_e(\alpha)} \frac{dx}{(e^{2x} - e^{-2x})}$ is :

Let $f : \mathbf{R} \to \mathbf{R}$ be a function such that $f(x) + 3f\left(\dfrac{\pi}{2} - x\right) = \sin x$, $x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ 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 __________.

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

The area of the region $\{(x, y) : 0 \leq y \leq 6-x,\ y^2 \geq 4x-3,\ x \geq 0\}$ is:

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

Let $A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix}$ be a singular matrix. Let $f(x) = \int_0^x \left(t^2 + 2t + 3\right) dt$, $x \in [1, \alpha]$. If M and m are respectively the maximum and the minimum values of $f$ in $[1, \alpha]$, then $3(M - m)$ is equal to :

The area of the region $\left\{(x, y) : x^2 - 8x \le y \le -x\right\}$ is :

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

The value of the integral $\int_0^2 \frac{\sqrt{x\left(x^2 + x + 1\right)}}{\left(\sqrt{x} + 1\right)\left(\sqrt{x^4 + x^2 + 1}\right)}\, dx$ is equal to :

Let $f : (1, \infty) \to \mathbb{R}$ be a function defined as $f(x) = \frac{x-1}{x+1}$. 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 :

If $\int_{\pi/6}^{\pi/4}\left(\cot\left(x - \frac{\pi}{3}\right)\cot\left(x + \frac{\pi}{3}\right) + 1\right)dx = \alpha \log_e\left(\sqrt{3} - 1\right)$, then $9\alpha^2$ is equal to __________.

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

Let $[\cdot]$ denote the greatest integer function. If $\int_0^{\mathrm{e}^3} \left[\frac{1}{\mathrm{e}^{x-1}}\right] \mathrm{d}x = \alpha - \log_\mathrm{e} 2$, then $\alpha^3$ is equal to __________.

$4\displaystyle\int_0^1\left(\frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right)\mathrm{d}x-3\log_e\left(\sqrt{3}\right)$ is equal to :

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 $I=\displaystyle\int_a^b \frac{9x^2}{1+5^x}\,\mathrm{d}x$ is equal to :

Let $f(x) = \int x^3 \sqrt{3 - x^2}\, dx$. If $5f\left(\sqrt{2}\right) = -4$, then $f(1)$ 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 $\int_0^{b-a}\left[x^2\right] dx = p - \sqrt{q} - \sqrt{r}$, $p, q, r \in \mathbb{N}$, $\gcd(p, q, r) = 1$, where $[\cdot]$ is the greatest integer function, then $p + q + r$ is equal to

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 __________

The integral $\displaystyle\int_0^{\pi} \dfrac{8x\,dx}{4\cos^2 x + \sin^2 x}$ is equal to

The area of the region $\left\{(x, y) : |x - y| \leq y \leq 4\sqrt{x}\right\}$ is

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

If the area of the region $\{(x, y) : |x - 5| \le y \le 4\sqrt{x}\}$ is $A$, then $3A$ 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 $\alpha=\displaystyle\int_1^2 f(x)\,\mathrm{d}x$, and $\beta=\displaystyle\int_1^2 g(x)\,\mathrm{d}x$, then the value of $9\alpha+\beta$ is :

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}:y^2=x-2$ 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 :

If $\displaystyle\int\dfrac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9}\,\mathrm{d}x=\dfrac{1}{\mathrm{m}}\left(\left(\sqrt{1+x^2}+x\right)^{\mathrm{n}}\left(\mathrm{n}\sqrt{1+x^2}-x\right)\right)+\mathrm{C}$ where C is the constant of integration and $\mathrm{m},\mathrm{n}\in\mathbf{N}$, then $\mathrm{m}+\mathrm{n}$ is equal to __________.

The integral $\int_0^{\pi} \frac{(x+3)\sin x}{1 + 3\cos^2 x}\, dx$ is equal to

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

If the area of the region $\{(x, y) : 1 + x^2 \leq y \leq \min\{x + 7, 11 - 3x\}\}$ is $A$, then $3A$ is equal to :

If $\int \left(\dfrac{1}{x} + \dfrac{1}{x^3}\right)\left(\sqrt[23]{3x^{-24} + x^{-26}}\right) \mathrm{d}x = -\dfrac{\alpha}{3(\alpha + 1)}\left(3x^{\beta} + x^{\gamma}\right)^{\frac{\alpha + 1}{\alpha}} + C$, $x > 0$, $(\alpha, \beta, \gamma \in \mathbf{Z})$, where $C$ is the constant of integration, then $\alpha + \beta + \gamma$ is equal to _________.

Let $f(x)$ be a positive function and $I_1 = \int\limits_{-\frac{1}{2}}^{1} 2x\, f\left(2x(1-2x)\right) dx$ and $I_2 = \int\limits_{-1}^{2} f\left(x(1-x)\right) dx$. Then the value of $\frac{I_2}{I_1}$ is equal to __________

The integral $\int\limits_{-1}^{\frac{3}{2}} \left(\left|\pi^2 x \sin(\pi x)\right|\right) dx$ 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 __________

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 $f''(x)-3af'(x)-f(x)=0$, $a>0$, then the area of the region $R=\{(x, y) \mid 0 \le y \le f(ax), 0 \le x \le 2\}$ is :

Let 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 $7I_1+12I_2$ 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 the function, $f(x)=\begin{cases}-3ax^2-2, & x<1 \\ a^2+bx, & x \geq 1\end{cases}$ 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 enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is :

If $\displaystyle\int e^x \left( \dfrac{x \sin^{-1} x}{\sqrt{1 - x^2}} + \dfrac{\sin^{-1} x}{\left(1 - x^2\right)^{3/2}} + \dfrac{x}{1 - x^2} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g\left(\dfrac{1}{2}\right)$ equals :

Let $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37) - I(24) = \frac{1}{4}\left(\frac{1}{b^{\frac{1}{13}}} - \frac{1}{c^{\frac{1}{13}}}\right)$, $b, c \in \mathbb{N}$, then $3(b + c)$ is equal to

The value of

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

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

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

Let $\int x^3 \sin x\, \mathrm{d}x = g(x) + C$, where C is the constant of integration. If $8\left(g\left(\dfrac{\pi}{2}\right) + g'\left(\dfrac{\pi}{2}\right)\right) = \alpha\pi^3 + \beta\pi^2 + \gamma$, $\alpha, \beta, \gamma \in \mathbf{Z}$, then $\alpha + \beta - \gamma$ equals :

If $\mathrm{I} = \displaystyle\int_0^{\frac{\pi}{2}} \dfrac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x}\, \mathrm{d}x$, then $\displaystyle\int_0^{2\mathrm{I}} \dfrac{x \sin x \cos x}{\sin^4 x + \cos^4 x}\, \mathrm{d}x$ equals :

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 $\{(x, y): x^2 + 4x + 2 \leq y \leq |x + 2|\}$ is equal to

The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is :

If $\displaystyle\int \dfrac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}}\, \mathrm{d}x = x\sqrt{x^2 + x + 1} + \alpha\sqrt{x^2 + x + 1} + \beta \log_e\left|x + \dfrac{1}{2} + \sqrt{x^2 + x + 1}\right| + \mathrm{C}$,

where C is the constant of integration, then $\alpha + 2\beta$ is equal to __________.

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

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

Let $f : \mathbf{R} \to \mathbf{R}$ be a twice differentiable function such that $f(2) = 1$. If $\mathrm{F}(x) = x f(x)$ for all $x \in \mathbf{R}$, $\int_0^2 x\, \mathrm{F}'(x)\, \mathrm{d}x = 6$ and $\int_0^2 x^2\, \mathrm{F}''(x)\, \mathrm{d}x = 40$, then $\mathrm{F}'(2) + \int_0^2 \mathrm{F}(x)\, \mathrm{d}x$ is equal to :

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x) = \int_0^x \mathrm{t}\, f(\mathrm{t})\, \mathrm{dt}$. If $g(x^3) = x^6 + x^7$, then value of $\sum_{r=1}^{15} f(\mathrm{r}^3)$ is :

If $f(x) = \int \frac{1}{x^{1/4}\left(1 + x^{1/4}\right)} \mathrm{d}x$, $f(0) = -6$, then $f(1)$ is equal to :

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