In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
Deduce that $$Z_n(h) = \sqrt{\frac{n}{2\mathrm{e}^{\beta}\pi\beta}} \int_{-\infty}^{+\infty} \left(\sum_{x \in \Lambda_n} \mathrm{e}^{(t+h)s_n(x)}\right) \mathrm{e}^{-\frac{nt^2}{2\beta}} \mathrm{~d}t$$

124. What is $\displaystyle\int_1^{16} [\sqrt{x}]\, dx$? (The symbol $[\,]$ denotes the floor function.)
(1) $30$ (2) $31$ (3) $32$ (4) $34$

124- What is the value of $\displaystyle\int_{0}^{2} \left[\frac{x}{2}\right] \frac{\sqrt{x}-1}{x}\, dx$?

• [(1)] $4 - 2\sqrt{2} - \ln 2$
• [(2)] $4 - 2\sqrt{2} + \ln 2$
• [(3)] $2 + \sqrt{2} - \ln 2$
• [(4)] $2 - \sqrt{2} + \ln 2$

123- What is the mean value of $f(x) = \dfrac{x^2 - 2}{x^2}$ on the interval $[2, 4]$?
(1) $\dfrac{5}{8}$ (2) $\dfrac{11}{16}$ (3) $\dfrac{3}{4}$ (4) $\dfrac{7}{8}$

124. The value of $\displaystyle\int_0^4 |1 - \sqrt{x}|\, dx$ is:
(1) $\dfrac{4}{3}$ (2) $\dfrac{5}{3}$ (3) $2$ (4) $3$

Show that $\int_1^n [u]([u]+1)f(u)\,du = 2\sum_{i=1}^{[n]} i \int_i^{i+1} f(u)\,du$ (or an equivalent integral identity involving the floor function).

Let $f$ be a differentiable function on $[0, 2\pi]$ with $f'(x)$ increasing. Show that $\int_0^{2\pi} f(x) \cos x \, dx \geq 0$.

Let $[ x ]$ denote the largest integer less than or equal to $x$. Then $\int_0^{n^{1/k}} \left[ x ^ { k } + n \right] dx$ equals
(a) $n ^ { 2 } + \sum_{i=1}^{n} i ^ { 1 / k }$
(b) $2 n ^ { ( 1 + k ) / k } - \sum_{i=1}^{n} i ^ { 1 / k }$
(c) $2 n ^ { ( 1 + k ) / k } - \sum_{i=1}^{n-1} i ^ { 1 / k }$
(d) None of these.

Let $f ( x ) = ( \tan x ) ^ { 3 / 2 } - 3 \tan x + \sqrt{\tan x}$. Consider the three integrals $I _ { 1 } = \int_0^1 f ( x ) \, dx$; $I _ { 2 } = \int_{0.3}^{1.3} f ( x ) \, dx$ and $I _ { 3 } = \int_{0.5}^{1.5} f ( x ) \, dx$. Then,
(a) $I _ { 1 } > I _ { 2 } > I _ { 3 }$
(b) $I _ { 2 } > I _ { 1 } > I _ { 3 }$
(c) $I _ { 3 } > I _ { 1 } > I _ { 2 }$
(d) $I _ { 1 } > I _ { 3 } > I _ { 2 }$

Find the integer part of $S = \displaystyle\sum_{k=2}^{9999} \dfrac{1}{\sqrt{k}}$.

The value of the integral $$\int _ { \pi / 2 } ^ { 5 \pi / 2 } \frac { e ^ { \tan ^ { - 1 } ( \sin x ) } } { e ^ { \tan ^ { - 1 } ( \sin x ) } + e ^ { \tan ^ { - 1 } ( \cos x ) } } d x$$ equals (A) 1 (B) $\pi$ (C) $e$ (D) none of these

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac{3}{4}$
(B) $\frac{4}{3}$
(C) $\frac{3}{2}$
(D) $\frac{2}{3}$

Evaluate $\displaystyle I = \int_{1/2014}^{2014} \frac{\tan^{-1} x}{x}\, dx$.
(A) $\dfrac{\pi}{2} \log(2014)$ (B) $\pi \log(2014)$ (C) $2\pi \log(2014)$ (D) $\dfrac{\pi}{4} \log(2014)$

Let $n$ be a positive integer. Define $$f ( x ) = \min \{ | x - 1 | , | x - 2 | , \ldots , | x - n | \}$$ Then $\int _ { 0 } ^ { n + 1 } f ( x ) d x$ equals
(A) $\frac { ( n + 4 ) } { 4 }$
(B) $\frac { ( n + 3 ) } { 4 }$
(C) $\frac { ( n + 2 ) } { 2 }$
(D) $\frac { ( n + 2 ) } { 4 }$

Let $n$ be a positive integer. Define $$f ( x ) = \min \{ | x - 1 | , | x - 2 | , \ldots , | x - n | \}$$ Then $\int _ { 0 } ^ { n + 1 } f ( x ) d x$ equals
(A) $\frac { ( n + 4 ) } { 4 }$
(B) $\frac { ( n + 3 ) } { 4 }$
(C) $\frac { ( n + 2 ) } { 2 }$
(D) $\frac { ( n + 2 ) } { 4 }$

The value of the integral $$\int _ { \pi / 2 } ^ { 5 \pi / 2 } \frac { e ^ { \tan ^ { - 1 } ( \sin x ) } } { e ^ { \tan ^ { - 1 } ( \sin x ) } + e ^ { \tan ^ { - 1 } ( \cos x ) } } d x$$ equals
(A) 1
(B) $\pi$
(C) $e$
(D) none of these

The value of the integral $$\int _ { \pi / 2 } ^ { 5 \pi / 2 } \frac { e ^ { \tan ^ { - 1 } ( \sin x ) } } { e ^ { \tan ^ { - 1 } ( \sin x ) } + e ^ { \tan ^ { - 1 } ( \cos x ) } } d x$$ equals
(A) 1
(B) $\pi$
(C) $e$
(D) none of these

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac{3}{4}$
(B) $\frac{4}{3}$
(C) $\frac{3}{2}$
(D) $\frac{2}{3}$

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $(2, 3)$
(D) is greater than 3

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $( 2, 3 )$
(D) is greater than 3

Let $f : [0,2] \rightarrow \mathbb{R}$ be a continuous function such that $$\frac{1}{2}\int_0^2 f(x)\,dx < f(2)$$ Then which of the following statements must be true?
(A) $f$ must be strictly increasing.
(B) $f$ must attain a maximum value at $x = 2$.
(C) $f$ cannot have a minimum at $x = 2$.
(D) None of the above.

Let $f : ( 0 , \infty ) \rightarrow \mathbb { R }$ be a continuous function such that for all $x \in ( 0 , \infty )$, $$f ( 2 x ) = f ( x )$$ Show that the function $g$ defined by the equation $$g ( x ) = \int _ { x } ^ { 2 x } f ( t ) \frac { d t } { t } \text { for } x > 0$$ is a constant function.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function such that $$f ( x + 1 ) = \frac { 1 } { 2 } f ( x ) \text { for all } x \in \mathbb { R } ,$$ and let $a _ { n } = \int _ { 0 } ^ { n } f ( x ) d x$ for all integers $n \geq 1$. Then:
(A) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $\int _ { 0 } ^ { 1 } f ( x ) d x$.
(B) $\lim _ { n \rightarrow \infty } a _ { n }$ does not exist.
(C) $\lim _ { n \rightarrow \infty } a _ { n }$ exists if and only if $\left| \int _ { 0 } ^ { 1 } f ( x ) d x \right| < 1$.
(D) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $2 \int _ { 0 } ^ { 1 } f ( x ) d x$.

For $n \in \mathbb { N }$, let $a _ { n }$ be defined as $$a _ { n } = \int _ { 0 } ^ { n } \frac { 1 } { 1 + n x ^ { 2 } } d x$$ Then $\lim _ { n \rightarrow \infty } a _ { n }$
(A) equals 0
(B) equals $\frac { \pi } { 4 }$
(C) equals $\frac { \pi } { 2 }$
(D) does not exist

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a twice differentiable one-to-one function. If $f ( 2 ) = 2 , f ( 3 ) = - 8$ and $$\int _ { 2 } ^ { 3 } f ( x ) d x = - 3$$ then $$\int _ { - 8 } ^ { 2 } f ^ { - 1 } ( x ) d x$$ equals
(A) $- 25$.
(B) $25$.
(C) $- 31$.
(D) $31$.