The integral $\int _ { 0 } ^ { 1 } \frac { 1 } { 7 ^ { \left[ \frac { 1 } { x } \right] } } d x$, where $[ \cdot ]$ denotes the greatest integer function, is equal to
(1) $1 - 6 \ln \left( \frac { 6 } { 7 } \right)$
(2) $1 + 6 \ln \left( \frac { 6 } { 7 } \right)$
(3) $1 - 7 \ln \left( \frac { 6 } { 7 } \right)$
(4) $1 + 7 \ln \left( \frac { 6 } { 7 } \right)$

$\int _ { 0 } ^ { 20 \pi } ( | \sin x | + | \cos x | ) ^ { 2 } \, d x$ is equal to:
(1) $10 \pi + 4$
(2) $10 \pi + 2$
(3) $20 \pi - 2$
(4) $20 \pi + 2$

Let $f ( x ) = \max \{ | x + 1 | , | x + 2 | , \ldots , | x + 5 | \}$. Then $\int _ { - 6 } ^ { 0 } f ( x ) \, dx$ is equal to $\_\_\_\_$.

The value of the integral $\frac { 48 } { \pi ^ { 4 } } \int _ { 0 } ^ { \pi } \left( \frac { 3 \pi x ^ { 2 } } { 2 } - x ^ { 3 } \right) \frac { \sin x } { 1 + \cos ^ { 2 } x } \, dx$ is equal to $\_\_\_\_$.

Let $f ( x ) = \min \{ [ x - 1 ] , [ x - 2 ] , \ldots , [ x - 10 ] \}$ where $[ t ]$ denotes the greatest integer $\leq t$. Then $\int _ { 0 } ^ { 10 } f ( x ) d x + \int _ { 0 } ^ { 10 } ( f ( x ) ) ^ { 2 } d x + \int _ { 0 } ^ { 10 } | f ( x ) | d x$ is equal to $\_\_\_\_$ .

The value of the integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \frac { \sin ( 6 x ) } { \sin x } d x$ is equal to $\_\_\_\_$.

Let f be a continuous function satisfying $\int _ { 0 } ^ { t ^ { 2 } } \left( f(x) + x^2 \right) dx = \frac { 4 } { 3 } t ^ { 3 } , \forall t > 0$. Then $f\left( \frac { \pi ^ { 2 } } { 4 } \right)$ is equal to
(1) $\pi ^ { 2 } \left( 1 - \frac { \pi ^ { 2 } } { 16 } \right)$
(2) $- \pi \left( 1 + \frac { \pi ^ { 3 } } { 16 } \right)$
(3) $\pi \left( 1 - \frac { \pi ^ { 3 } } { 16 } \right)$
(4) $- \pi ^ { 2 } \left( 1 + \frac { \pi ^ { 2 } } { 16 } \right)$

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f\left(\frac{\pi}{4}\right) = \sqrt{2}$, $f\left(\frac{\pi}{2}\right) = 0$ and $f'\left(\frac{\pi}{2}\right) = 1$ and let $g(x) = \int_x^{\pi/4} (f'(t)\sec t + \tan t \cdot f(t)\sec t)\,dt$. Then $\lim_{x \to \pi/2} \frac{g(x)}{(x - \pi/2)^2}$ is equal to $\_\_\_\_$.

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 $\_\_\_\_$.

Let $[ \mathrm { x } ]$ denote the greatest integer $\leq \mathrm { x }$. Consider the function $\mathrm { f } ( \mathrm { x } ) = \max \left\{ \mathrm { x } ^ { 2 } , 1 + [ x ] \right\}$. Then the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ is :
(1) $\frac { 5 + 4 \sqrt { 2 } } { 3 }$
(2) $\frac { 8 + 4 \sqrt { 2 } } { 3 }$
(3) $\frac { 1 + 5 \sqrt { 2 } } { 3 }$
(4) $\frac { 4 + 5 \sqrt { 2 } } { 3 }$

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

The minimum value of the function $f ( x ) = \int _ { 0 } ^ { 2 } e ^ { | x - t | } d t$ is
(1) $2 ( e - 1 )$
(2) $2 e - 1$
(3) 2
(4) $e ( e - 1 )$

Let $[ t ]$ denote the greatest integer $\leq t$. Then $\frac { 2 } { \pi } \int _ { \frac { \pi } { 6 } } ^ { \frac { 5 \pi } { 6 } } ( 8 [ \operatorname { cosec } x ] - 5 [ \cot x ] ) d x$ is equal to $\_\_\_\_$

$\lim _ { x \rightarrow \frac { \pi } { 2 } } \left( \frac { 1 } { \left( x - \frac { \pi } { 2 } \right) ^ { 2 } } \int _ { x ^ { 3 } } ^ { \left( \frac { \pi } { 2 } \right) ^ { 3 } } \cos \left( \frac { 1 } { t ^ { 3 } } \right) d t \right)$ is equal to
(1) $\frac { 3 \pi } { 8 }$
(2) $\frac { 3 \pi ^ { 2 } } { 4 }$
(3) $\frac { 3 \pi ^ { 2 } } { 8 }$
(4) $\frac { 3 \pi } { 4 }$

Let $f: R \rightarrow R$ be a function defined $f(x) = \frac{x}{(1 + x^4)^{1/4}}$ and $g(x) = f(f(f(f(x))))$, then $18\int_0^{\sqrt{2\sqrt{5}}} x^2 g(x)\, dx$
(1) 33
(2) 36
(3) 42
(4) 39

Let $a$ and $b$ be real constants such that the function $f$ defined by $$f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1 \\ bx + 2, & x > 1 \end{cases}$$ be differentiable on $R$. Then, the value of $\int_{-2}^{2} f(x)\, dx$ equals
(1) $\frac{15}{6}$
(2) $\frac{19}{6}$
(3) 21
(4) 17

The value of $\int_0^1 (2x^3 - 3x^2 - x + 1)^{\frac{1}{3}} dx$ is equal to:
(1) 0
(2) 1
(3) 2
(4) $-1$

If the value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \left( \frac { x ^ { 2 } \cos x } { 1 + \pi ^ { x } } + \frac { 1 + \sin ^ { 2 } x } { 1 + e ^ { ( \sin x ) ^ { 2023 } } } \right) d x = \frac { \pi } { 4 } ( \pi + a ) - 2$, then the value of $a$ is
(1) 3
(2) $- \frac { 3 } { 2 }$
(3) 2
(4) $\frac { 3 } { 2 }$

Let $\beta ( \mathrm { m } , \mathrm { n } ) = \int _ { 0 } ^ { 1 } x ^ { \mathrm { m } - 1 } ( 1 - x ) ^ { \mathrm { n } - 1 } \mathrm {~d} x , \mathrm {~m} , \mathrm { n } > 0$. If $\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 10 } \right) ^ { 20 } \mathrm {~d} x = \mathrm { a } \times \beta ( \mathrm { b } , \mathrm { c } )$, then $100 ( \mathrm { a } + \mathrm { b } + \mathrm { c } )$ equals
(1) 1021
(2) 2120
(3) 2012
(4) 1120

Let $\int _ { \alpha } ^ { \log _ { e } 4 } \frac { \mathrm {~d} x } { \sqrt { \mathrm { e } ^ { x } - 1 } } = \frac { \pi } { 6 }$. Then $\mathrm { e } ^ { \alpha }$ and $\mathrm { e } ^ { - \alpha }$ are the roots of the equation : (1) $x ^ { 2 } + 2 x - 8 = 0$ (2) $x ^ { 2 } - 2 x - 8 = 0$ (3) $2 x ^ { 2 } - 5 x + 2 = 0$ (4) $2 x ^ { 2 } - 5 x - 2 = 0$

If $\int_0^{\frac{\pi}{3}} \cos^4 x \, dx = a\pi + b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a + 8b$ is equal to:
(1) 2
(2) 1
(3) 3
(4) $\frac{3}{2}$

For $0 < \mathrm { a } < 1$, the value of the integral $\int _ { 0 } ^ { \pi } \frac { d x } { 1 - 2 a \cos x + a ^ { 2 } }$ is :
(1) $\frac { \pi ^ { 2 } } { \pi + a ^ { 2 } }$
(2) $\frac { \pi ^ { 2 } } { \pi - a ^ { 2 } }$
(3) $\frac { \pi } { 1 - a ^ { 2 } }$
(4) $\frac { \pi } { 1 + a ^ { 2 } }$

Let $y = f(x)$ be a thrice differentiable function on $(-5, 5)$. Let the tangents to the curve $y = f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x$-axis. If $27\int_1^3 \left(f'(t)\right)^2 + 1\right) f''(t)\, dt = \alpha + \beta\sqrt{3}$ where $\alpha, \beta$ are integers, then the value of $\alpha + \beta$ equals
(1) $-14$
(2) 26
(3) $-16$
(4) 36

If the value of the integral $\int _ { - 1 } ^ { 1 } \frac { \cos \alpha x } { 1 + 3 ^ { x } } d x$ is $\frac { 2 } { \pi }$. Then, a value of $\alpha$ is
(1) $\frac { \pi } { 3 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$

Let $f: R \rightarrow R$ be defined $f(x) = ae^{2x} + be^x + cx$. If $f(0) = -1$, $f'(\log_e 2) = 21$ and $\int_0^{\log 4} (f(x) - cx)\, dx = \frac{39}{2}$, then the value of $|a + b + c|$ equals:
(1) 16
(2) 10
(3) 12
(4) 8