Let $I\left(x\right)=\int \frac{x^2(xse{c}^{2}x+\tan x)}{{(x\tan x+1)}^2}dx$. If $I\left(0\right)=0,$ then $I\left(\frac{\mathrm{\pi }}{4}\right)$ is equal to

(a) ${\log }_e\frac{{(\mathrm{\pi }+4)}^2}{16}+\frac{{\mathrm{\pi }}^2}{4(\mathrm{\pi }+4)}$                        (b) ${\log }_e\frac{{(\mathrm{\pi }+4)}^2}{32}+\frac{{\mathrm{\pi }}^2}{4(\mathrm{\pi }+4)}$

(c) ${\log }_e\frac{{(\mathrm{\pi }+4)}^2}{16}-\frac{{\mathrm{\pi }}^2}{4(\mathrm{\pi }+4)}$                        (d) ${\log }_e\frac{{(\mathrm{\pi }+4)}^2}{32}-\frac{{\mathrm{\pi }}^2}{4(\mathrm{\pi }+4)}$

Let $I\left(x\right)=\int \frac{(x+1)}{x{(1+x{e}^x)}^2}dx,x>0. If \underset{x\to \infty }{\lim }I\left(x\right)=0, then I\left(1\right) is equal to$

(a) $\frac{e+1}{e+2}-{\log }_e(e+1)$                            (b) $\frac{e+2}{e+1}-{\log }_e(e+1)$

(c) $\frac{e+2}{e+1}+{\log }_e(e+1)$                             (d) $\frac{e+1}{e+2}+{\log }_e(e+1)$

For $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\mathbb{N}},$ if $\int ({\left(\frac{x}{e}\right)}^{2x}+{\left(\frac{e}{x}\right)}^{2x}){\log }_{e}xdx=\frac{1}{\alpha }{\left(\frac{x}{e}\right)}^{\beta x}-\frac{1}{\gamma }{\left(\frac{e}{x}\right)}^{\delta x}+C,$ where $e=\sum _{n=0}^{\infty }\frac{1}{n!}$

and C is constant of integration, then $\alpha +2\beta +3\gamma -4\delta$ is equal to

(a) 1                          (b) - 8                          (c) 4                     (d) - 4

 $$\begin{gathered} Let f\left(x\right)=\int \frac{2x}{(x^2+1)(x^2+3)}dx. \\ If f\left(3\right)=\frac{1}{2}({\log }_{e}5-{\log }_{e}6), then f\left(4\right) is equal to \end{gathered}$$

(a) ${\log }_{e}17-{\log }_{e}18$                          (b) ${\log }_{e}19-{\log }_{e}20$

(c) $\frac{1}{2}({\log }_{e}19-{\log }_{e}17)$                  (d) $\frac{1}{2}({\log }_{e}17-{\log }_{e}19)$

Let $I\left(x\right)=\int \sqrt{\frac{x+7}{x}}dx$ and $I\left(9\right)=12+7{\log }_{e}7.$ If $I\left(1\right)=\alpha +7{\log }_e(1+2\sqrt{2})$ then ${\alpha }^4$ is equal to ______ .

Let $f\left(x\right)=\int \frac{dx}{(3+4x^2)\sqrt{4-3x^2}},\left|x\right|<\frac{2}{\sqrt{3}}.$ If $f\left(0\right)=0$ and $f\left(1\right)=\frac{1}{\alpha \beta }{\tan }^{-1}\left(\frac{\alpha }{\beta }\right),\alpha ,\beta >0,$ then ${\alpha }^2+{\beta }^2$ is equal to __________ .

If $\int \sqrt{sec2x-1}dx=\alpha {\log }_e|\cos 2x+\beta +\sqrt{\cos 2x(1+\cos \frac{1}{\beta }x)}|$ + constant, then $\beta -\alpha$ is equal to _________ .

Let $5f\left(x\right)+4f\left(\frac{1}{x}\right)=\frac{1}{x}+3,x>0.$ Then $18{\int }_1^{2}f\left(x\right)dx$ is equal to

(a) $5{\log }_{e}2+3$          (b) $10{\log }_{e}2+6$          (c) $5{\log }_{e}2-3$           (d) $10{\log }_{e}2-6$

Let $f\left(x\right)$ be a function satisfying $f\left(x\right)+f(\mathrm{\pi }-\mathrm{x})={\mathrm{\pi }}^2\forall \mathrm{x}\in \mathrm{ℝ}.$ Then ${\int }_0^{\pi }f\left(x\right)\sin x dx$ is equal to

(a) ${\mathrm{\pi }}^2$                   (b) $2{\mathrm{\pi }}^2$                    (c) $\frac{{\mathrm{\pi }}^2}{2}$                     (d) $\frac{{\mathrm{\pi }}^2}{4}$

If $I\left(x\right)=\int e^{{\sin }^{2}x}(\cos x\sin 2x-\sin x)dx$ and $I\left(0\right)=1,$ then $I\left(\frac{\mathrm{\pi }}{3}\right)$ is equal to

(a) $-e^{\frac{3}{4}}$                   (b) $e^{\frac{3}{4}}$                  (c) $\frac{1}{2}{e}^{\frac{3}{4}}$                 (d) $-\frac{1}{2}{e}^{\frac{3}{4}}$

Let $f$ be a continuous function satisfying ${\int }_0^{t^2}\left(f\right(x)+x^2)dx=\frac{4}{3}{t}^3, \forall t>0.$ Then $f\left(\frac{{\mathrm{\pi }}^2}{4}\right)$ is equal to

(a) $\mathrm{\pi }(1-\frac{{\mathrm{\pi }}^3}{16})$               (b) $-{\mathrm{\pi }}^2(1+\frac{{\mathrm{\pi }}^3}{16})$             (c) $-\mathrm{\pi }(1+\frac{{\mathrm{\pi }}^3}{16})$                  (d) ${\mathrm{\pi }}^2(1-\frac{{\mathrm{\pi }}^2}{16})$

The value of the integral ${\int }_{-{\log }_{e}2}^{{\log }_{e}2}{e}^x\left({\log }_e\right(e^x+\sqrt{1+e^{2x}}\left)\right)dx$ is equal to

(a) ${\log }_e\left(\frac{2(2+\sqrt{5})}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$                        (b) ${\log }_e\left(\frac{\sqrt{2}{(3-\sqrt{5})}^2}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$

(c) ${\log }_e\left(\frac{{(2+\sqrt{5})}^2}{\sqrt{1+\sqrt{5}}}\right)+\frac{\sqrt{5}}{2}$                        (d) ${\log }_e\left(\frac{\sqrt{2}{(2+\sqrt{5})}^2}{\sqrt{1+\sqrt{5}}}\right)-\frac{\sqrt{5}}{2}$

If $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a continuous function satisfying ${\int }_0^{\mathrm{\pi }/2}f(\sin 2x)\sin xdx+\alpha$ ${\int }_0^{\mathrm{\pi }/4}f\left(\cos 2x\right)\cos xdx=0,$ then the value of $\alpha$ is

(a) $-\sqrt{2}$                 (b) $\sqrt{3}$                    (c) $-\sqrt{3}$                (d) $\sqrt{2}$

Let the function $f:[0,2]\to \mathrm{ℝ}$ be defined as $f\left(x\right)=\{\begin{array}{ll}{e}^{min\{x^2,x-[x\left]\right\}},& x\in [0,1)\\ e^{[x-{\log }_{e}x]},& x\in [1,2)\end{array}$

Where [t] denotes the greatest integer less than or equal to t. Then the value of the integral ${\int }_0^{2}xf\left(x\right)dx$ is

(a) $(e-1)(e^2+\frac{1}{2})$              (b) $2e-\frac{1}{2}$              (c) $1+\frac{3e}{2}$                (d) $2e-1$

${\int }_0^{\infty }\frac{6}{e^{3x}+6e^{2x}+11e^x+6}dx=$

(a) ${\log }_e\left(\frac{64}{27}\right)$                  (b) ${\log }_e\left(\frac{512}{81}\right)$                    (c) ${\log }_e\left(\frac{256}{81}\right)$                  (d) ${\log }_e\left(\frac{32}{27}\right)$

The value of $\frac{e^{-{\displaystyle \frac{\mathrm{\pi }}{4}}}+{\int }_0^{{\displaystyle \frac{\mathrm{\pi }}{4}}}{e}^{-x}{\tan }^{50}xdx}{{\int }_0^{{\displaystyle \frac{\mathrm{\pi }}{4}}}{e}^{-x}({\tan }^{49}x+{\tan }^{51}x)dx}$ is

(a) 51                    (b) 49                  (c) 25                     (d) 50

If ${\int }_0^1\frac{1}{(5+2x-2x^2)(1+e^{(2-4x)})}dx=\frac{1}{\alpha }{\log }_e\left(\frac{\alpha +1}{\beta }\right), \alpha ,\beta >0,$ then ${\alpha }^4-{\beta }^4$ is equal to

(a) - 21                       (b) 0                      (c) 21                         (d) 19

$\underset{n\to \infty }{\lim }[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+...+\frac{1}{2n}]$ is equal to

(a) ${\log }_{e}2$                    (b) ${\log }_e\left(\frac{2}{3}\right)$               (c) 0                    (d) ${\log }_e\left(\frac{3}{2}\right)$

The value of the integral ${\int }_{-\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{4}}\frac{x+{\displaystyle \frac{\mathrm{\pi }}{4}}}{2-\cos 2x}dx$ is

(a) $\frac{{\mathrm{\pi }}^2}{12\sqrt{3}}$                   (b) $\frac{{\mathrm{\pi }}^2}{6}$                    (c) $\frac{{\mathrm{\pi }}^2}{3\sqrt{3}}$                 (d) $\frac{{\mathrm{\pi }}^2}{6\sqrt{3}}$

${\int }_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}}\frac{48}{\sqrt{9-4x^2}}dx is equal to$

(a) $2\mathrm{\pi }$                       (b) $\frac{\mathrm{\pi }}{6}$                        (c) $\frac{\mathrm{\pi }}{3}$                       (d) $\frac{\mathrm{\pi }}{2}$

The minimum value of the function $f\left(x\right)={\int }_0^2{e}^{|x-t|}dt$ is

(a) 2                      (b) 2(e - 1)                   (c) 2e - 1              (d) e(e - 1)

The integral $16{\int }_1^2\frac{dx}{x^3{(x^2+2)}^2}$ is equal to

(a) $\frac{11}{6}-{\log }_{e}4$              (b) $\frac{11}{12}+{\log }_{e}4$               (c) $\frac{11}{6}+{\log }_{e}4$              (d) $\frac{11}{12}-{\log }_{e}4$

Let [$x$] denote the greatest integer $\le x$. Consider the function $f\left(x\right)=max\{x^2,1+[x\left]\right\}.$ Then the value of the integral ${\int }_0^{2}f\left(x\right)dx$ is

(a) $\frac{5+4\sqrt{2}}{3}$                (b) $\frac{8+4\sqrt{2}}{3}$             (c) $\frac{1+5\sqrt{2}}{3}$                  (d) $\frac{4+5\sqrt{2}}{3}$

Let $f\left(x\right)=x+\frac{a}{{\mathrm{\pi }}^2-4}\sin x+\frac{b}{{\mathrm{\pi }}^2-4}\cos x,x\in \mathrm{ℝ}$ be a function which satisfies

$f\left(x\right)=x+{\int }_0^{\mathrm{\pi }/2}\sin (x+y)f\left(y\right)dy.$ Then $(a+b)$ is equal to

(a) $-\mathrm{\pi }(\mathrm{\pi }+2)$                   (b) $-\mathrm{\pi }(\mathrm{\pi }-2)$                  (c) $-2\mathrm{\pi }(\mathrm{\pi }+2)$                   (d) $-2\mathrm{\pi }(\mathrm{\pi }-2)$

The value of the integral ${\int }_{1/2}^2\frac{{\tan }^{-1}x}{x}dx$ is equal to

(a) $\frac{1}{2}{\log }_{e}2$                     (b) ${\mathrm{\pi log}}_{e}2$                   (c) $\frac{\mathrm{\pi }}{4}{\log }_{e}2$                 (d) $\frac{\mathrm{\pi }}{2}{\log }_{e}2$

The value of the integral ${\int }_1^2\left(\frac{t^4+1}{t^6+1}\right)dt$ is

(a) ${\tan }^{-1}2-\frac{1}{3}{\tan }^{-1}8+\frac{\mathrm{\pi }}{3}$                        (b) ${\tan }^{-1}2+\frac{1}{3}{\tan }^{-1}8-\frac{\mathrm{\pi }}{3}$

(c) ${\tan }^{-1}\frac{1}{2}+\frac{1}{3}{\tan }^{-1}8-\frac{\mathrm{\pi }}{3}$                      (d) ${\tan }^{-1}\frac{1}{2}-\frac{1}{3}{\tan }^{-1}8+\frac{\mathrm{\pi }}{3}$

If [$t$] denotes the greatest integer $\le t,$ then the value of $\frac{3(e-1)}{e}{\int }_1^2{x}^2{e}^{\left[x\right]+\left[x^3\right]}dx$ is

(a) $e^8-1$                (b) $e^7-1$                (c) $e^9-e$                (d) $e^8-e$

$\underset{n\to \infty }{\lim }\frac{3}{n}\{4+{(2+\frac{1}{n})}^2+{(2+\frac{2}{n})}^2+...+{(3-\frac{1}{n})}^2\}$ is equal to

(a) $\frac{19}{3}$                           (b) 12                               (c) 0                                 (d) 19

Let $\alpha \in (0,1)$ and $\beta ={\log }_e(1-\alpha )$.

Let $P_n\left(x\right)=x+\frac{x^2}{2}+\frac{x^3}{3}+...+\frac{x^n}{n},x\in (0, 1).$

Then the integral ${\int }_0^{\alpha }\frac{t^{50}}{1-t}dt$ is equal to

(a) $P_{50}\left(\alpha \right)-\beta$                 (b) $-(\beta +P_{50}(\alpha \left)\right)$                  (c) $\beta +P_{50}\left(\alpha \right)$                   (d) $\beta -P_{50}\left(\alpha \right)$

The value of ${\int }_{\frac{\mathrm{\pi }}{3}}^{\frac{\mathrm{\pi }}{2}}\frac{(2+3\sin x)}{\sin x(1+\cos x)}dx$ is equal to

(a) $\frac{7}{2}-\sqrt{3}-{\log }_e\sqrt{3}$                           (b) $\frac{10}{3}-\sqrt{3}-{\log }_e\sqrt{3}$

(c) $\frac{10}{3}-\sqrt{3}+{\log }_e\sqrt{3}$                        (d) $-2+3\sqrt{3}+{\log }_e\sqrt{3}$

If $\varphi \left(x\right)=\frac{1}{\sqrt{x}}{\int }_{\frac{\mathrm{\pi }}{4}}^x(4\sqrt{2}\sin t-3\varphi \text{'}(t\left)\right)dt,x>0,$ then $\varphi \text{'}\left(\frac{\mathrm{\pi }}{4}\right)$ is equal to

(a) $\frac{4}{6+\sqrt{\mathrm{\pi }}}$                      (b) $\frac{8}{6+\sqrt{\mathrm{\pi }}}$                  (c) $\frac{8}{\sqrt{\mathrm{\pi }}}$                   (d) $\frac{4}{6-\sqrt{\mathrm{\pi }}}$

Let $\alpha >0.$ If ${\int }_0^{\alpha }\frac{x}{\sqrt{x+\alpha }-\sqrt{x}}dx=\frac{16+20\sqrt{2}}{15},$ Then $\alpha$ is equal to

(a) 2                       (b) $2\sqrt{2}$                           (c) 4                      (d) $\sqrt{2}$

Let $f\left(x\right)=\frac{x}{{(1+x^n)}^{{\displaystyle \frac{1}{n}}}},x\in \mathrm{ℝ}-\{-1\},n\in \mathrm{\mathbb{N}},n>2.$ If $f^n\left(x\right)=(fofof .... upto n times) \left(x\right)$, then $\underset{n\to \infty }{\lim }{\int }_0^1{x}^{n-2}\left(f^n\right(x\left)\right)dx$ is equal to __________.

Let $\left[t\right]$ denotes the greatest integer $\le f.$ Then $\frac{2}{\mathrm{\pi }}{\int }_{\mathrm{\pi }/6}^{5\mathrm{\pi }/6}\left(8\right[\cos ecx]-5[cotx\left]\right)dx$ is equal to ________.

Let $\left[t\right]$ denote the greatest integer function. If ${\int }_0^{2·4}\left[x^2\right]dx=\alpha +\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$, then $\alpha +\beta +\gamma +\delta$ is equal to ________ .

For $m,n>0,$ let $\alpha (m,n)={\int }_0^2{t}^m{(1+3t)}^{n}dt.$ If $11\alpha (10,6)+18\alpha (11,5)=p{\left(14\right)}^6,$ then $p$ is equal to ________ .

If ${\int }_{-0.15}^{0.15}|100x^2-1|dx=\frac{k}{3000},$ then $k$ is equal to _______ .

$$\begin{gathered} Let for x\in \mathrm{ℝ},S_0\left(x\right)=x,S_k\left(x\right)=C_{k}x+k{\int }_0^x{S}_{k-1}\left(t\right)dt, where C_0=1, C_k=1-{\int }_0^1{S}_{k-1}\left(x\right)dx,k=1,2,3,... \\ Then S_2\left(3\right)+6C_3 is equal to \_\_\_\_\_\_\_\_\_\_. \end{gathered}$$ 

Let 

$f_n={\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\sum _{k=1}^n{\sin }^{k-1}x\right)\left(\sum _{k=1}^n\right(2k-1\left){\sin }^{k-1}x\right)\cos xdx,n\in \mathrm{\mathbb{N}}.$

Then $f_{21}-f_{20}$ is equal to ____________ .

If ${\int }_0^1(x^{21}+x^{14}+x^7){(2x^{14}+3x^7+6)}^{1/7}dx=\frac{1}{l}{\left(11\right)}^m where l,m,n\in \mathrm{\mathbb{N}},m$ and $n$ are coprime, then $l+m+n$ is equal to _________ .

If ${\int }_0^{\mathrm{\pi }}\frac{5^{\cos x}(1+\cos x\cos 3x+{\cos }^{2}x+{\cos }^{3}x\cos 3x)dx}{1+5^{\cos x}}=\frac{k\mathrm{\pi }}{16},$ then $k$ is equal to ____________ .

The value of $\frac{8}{\mathrm{\pi }}{\int }_0^{\mathrm{\pi }/2}\frac{{\left(\cos x\right)}^{2023}}{{\left(\sin x\right)}^{2023}+{\left(\cos x\right)}^{2023}}dx$ is _________ .

The value of $12{\int }_0^3|x^2-3x+2|dx$ is _______ .

If ${\int }_{\frac{1}{3}}^3\left|{\log }_{e}x\right|dx=\frac{m}{n}{\log }_e\left(\frac{n^2}{e}\right),$ where $m$ and  $n$ are coprime natural numbers, then $m^2+n^2-5$ is equal to ______ .

$\underset{x\to 0}{\lim }\frac{48}{x^4}{\int }_0^x\frac{t^3}{t^6+1}dt$ is equal to ________ .

The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to

(a) 6                       (b) 3                             (c) 5                         (d) 4

The area of the region $\left\{\right(x,y):x^2\le y\le 8-x^2,y\le 7\}$ is

(a) 24                           (b) 20                              (c) 18                            (d) 21

Area of the region $\left\{\right(x,y):x^2+{(y-2)}^2\le 4,x^2\ge 2y\}$ is

(a) $2\mathrm{\pi }-\frac{16}{3}$                    (b) $\mathrm{\pi }-\frac{8}{3}$                    (c) $\mathrm{\pi }+\frac{8}{3}$                    (d) $2\mathrm{\pi }+\frac{16}{3}$

The area of the region enclosed by the curve $y=x^3$ and its tangent at the point $(-1,-1)$ is

(a) $\frac{27}{4}$                       (b) $\frac{31}{4}$                        (c) $\frac{23}{4}$                         (d) $\frac{19}{4}$

The area of the region enclosed by the curve $f\left(x\right)=max\{\sin x,\cos x\},-\mathrm{\pi }\le \mathrm{x}\le \mathrm{\pi }$ and the x-axis is

(a) $4\left(\sqrt{2}\right)$                     (b) $2(\sqrt{2}+1)$                    (c) $2\sqrt{2}(\sqrt{2}+1)$                         (d) 4

The area of the region $\left\{\right(x,y):x^2\le y\le |x^2-4|,y\ge 1\}$ is

(a) $\frac{4}{3}(4\sqrt{2}+1)$                   (b) $\frac{3}{4}(4\sqrt{2}-1)$                  (c) $\frac{3}{4}(4\sqrt{2}+1)$                   (d) $\frac{4}{3}(4\sqrt{2}-1)$

The area of the region given by $\left\{\right(x,y):xy\le 8,1\le y\le x^2\}$ is

(a) $16{\log }_{e}2-\frac{14}{3}$                   (b) $8{\log }_{e}2-\frac{13}{3}$                    (c) $8{\log }_{e}2+\frac{7}{6}$                   (d) $16{\log }_{e}2+\frac{7}{3}$

The area enclosed by the curves $y^2+4x=4$ and $y-2x=2$ is

(a) $\frac{23}{3}$                           (b) $\frac{22}{3}$                                    (c) $9$                                (d) $\frac{25}{3}$

Let $A=\left\{\right(x,y)\in {\mathrm{ℝ}}^2:y\ge 0,2x\le y\le \sqrt{4-{(x-1)}^2}\}$ and  $B=\left\{\right(x,y)\in \mathrm{ℝ}\times \mathrm{ℝ}:0\le y\le min\{2x,\sqrt{4-{(x-1)}^2}\left\}\right\}$.

Then the ratio of the area of A to the area of B is

(a) $\frac{\mathrm{\pi }}{\mathrm{\pi }+1}$                        (b) $\frac{\mathrm{\pi }-1}{\mathrm{\pi }+1}$                           (c) $\frac{\mathrm{\pi }+1}{\mathrm{\pi }-1}$                            (d) $\frac{\mathrm{\pi }}{\mathrm{\pi }-1}$

Let $∆$be the area of the region $\left\{\right(x,y)\in {\mathrm{ℝ}}^2:x^2+y^2\le 21,y^2\le 4x,x\ge 1\}.$ Then $\frac{1}{2}(∆-21{\sin }^{-1}\frac{2}{\sqrt{7}})$ is equal to

(a) $\sqrt{3}-\frac{2}{3}$                    (b) $2\sqrt{3}-\frac{2}{3}$                              (c) $2\sqrt{3}-\frac{1}{3}$                              (d) $\sqrt{3}-\frac{4}{3}$

The area of the region  $A=\left\{\right(x,y):|\cos x-\sin x|\le y\le \sin x,0\le x\le \frac{\mathrm{\pi }}{2}\} is$

(a) $\sqrt{5}-2\sqrt{2}+1$                  (b) $\sqrt{5}+2\sqrt{2}-4.5$                    (c) $1-\frac{3}{\sqrt{2}}+\frac{4}{\sqrt{5}}$                        (d) $\frac{3}{\sqrt{5}}-\frac{3}{\sqrt{2}}+1$

Let $q$ be the maximum integral value of $p$ in [0, 10] for which the roots of the equation $x^2-px+\frac{5}{4}p=0$ are rational.

Then the area of the region $\left\{\right(x,y):0\le y\le {(x-q)}^2,0\le x\le q\}$ is

(a) 164                             (b) 243                             (c) $\frac{125}{3}$                            (d) 25

If the area of region $S=\left\{\right(x,y):2y-y^2\le x^2\le 2y,x\ge y\}$ is equal to $\frac{n+2}{n+1}-\frac{\mathrm{\pi }}{n-1},$ then the natural number $n$ is equal to _______ .

Let the area enclosed by the lines $x+y=2,y=0,x=0$ and the curve $f\left(x\right)=min\{x^2+\frac{3}{4},1+[x\left]\right\}$ where $\left[x\right]$ denotes the greatest integer $\le x$, be A. Then the value of 12A is ________ .

Let $y=p\left(x\right)$ be the parabola passing through the points $(-1,0),(0,1)$ and $(1,0)$. If the area of the region $\left\{\right(x,y):{(x+1)}^2+{(y-1)}^2\le 1,y\le p(x\left)\right\}$ is A, then $12(\mathrm{\pi }-4\mathrm{A})$ is equal to __________ .

If the area of the region $\left\{\right(x,y):|x^2-2|\le y\le x\}$ is A, then $6A+16\sqrt{2}$ is equal to ______ .

If A is the area in the first quadrant enclosed by the curve $C:2x^2-y+1=0,$ the tangent to C at the point (1, 3) and the line $x+y=1,$ then the value of 60A is ________ .

If the area bounded by the curve $2y^2=3x,$ lines $x+y=3,y=0$and outside the circle ${(x-3)}^2+y^2=2$ is A, then $4(\mathrm{\pi }+4\mathrm{A})$ is equal to ________ .

Let A be the area bounded by the curve $y=x|x-3|,$ the x-axis and the ordinates $x=-1$and $x=2$. Then 12A is equal to _______ .

If the area of the region bounded by the curves $y^2-2y=-x,x+y=0$ is A, then 8 A is equal to _______ .

If the area enclosed by the parabolas $P_1:2y=5x^2$ and $P_2:x^2-y+6=0$ is equal to the area enclosed by $P_1$ and $y=\alpha x,\alpha >0,$ then ${\alpha }^3$ is equal to ____________ .

Let $\alpha$ be the area of the larger region bounded by the curve $y^2=8x$ and the lines $y=x$ and $x=2,$ which lies in the first quadrant. Then the value of $3\alpha$ is equal to __________ .

Let A be the area of the region $\left\{\right(x,y):y\ge x^2,y\ge {(1-x)}^2,y\le 2x(1-x\left)\right\}$. Then 540 A is equal to _________ .

Let for $x\in R,f\left(x\right)=\frac{x+\left|x\right|}{2}$ and $g\left(x\right)=\{\begin{array}{ll}x,& x<0\\ x^2,& x\ge 0\end{array}.$  Then area bounded by the curve $y=\left(fog\right)\left(x\right)$ and the lines $y=0,2y-x=15$ is equal to ________ .

Let the area of the region $\left\{\right(x,y):|2x-1|\le y\le |x^2-x|,0\le x\le 1\}$ be A. Then ${(6A+11)}^2$ is equal to ________ .

The set $A=\{x:x\in R, x^2=16 and 2x=6\} equals$