41. Given $f ( x ) = \left\{ \begin{array} { l l } x + 1 & \text { for } x < 0 , \\ \cos \pi x & \text { for } x \geq 0 , \end{array} \quad \int _ { - 1 } ^ { 1 } f ( x ) d x = \right.$
(A) $\frac { 1 } { 2 } + \frac { 1 } { \pi }$
(B) $- \frac { 1 } { 2 }$
(C) $\frac { 1 } { 2 } - \frac { 1 } { \pi }$
(D) $\frac { 1 } { 2 }$
(E) $\quad - \frac { 1 } { 2 } + \pi$ [Figure]

The function $f$ is defined by $f ( x ) = \left\{ \begin{array} { l l } 2 & \text { for } x < 3 \\ x - 1 & \text { for } x \geq 3 \end{array} \right.$. What is the value of $\int _ { 1 } ^ { 5 } f ( x ) d x$ ?
(A) 2
(B) 6
(C) 8
(D) 10
(E) 12

The graph of the piecewise linear function $f$ is shown above. What is the value of $\int _ { - 1 } ^ { 9 } ( 3 f ( x ) + 2 ) d x$ ?
(A) 7.5
(B) 9.5
(C) 27.5
(D) 47
(E) 48.5

Calculate the following integrals whenever possible. If a given integral does not exist, state so. Note that $[ x ]$ denotes the integer part of $x$, i.e., the unique integer $n$ such that $n \leq x < n + 1$. a) $\int _ { 1 } ^ { 4 } x ^ { 2 } d x$
Answer: $\_\_\_\_$ b) $\int _ { 1 } ^ { 3 } [ x ] ^ { 2 } d x$
Answer: $\_\_\_\_$ c) $\int _ { 1 } ^ { 2 } \left[ x ^ { 2 } \right] d x$
Answer: $\_\_\_\_$ d) $\int _ { - 1 } ^ { 1 } \frac { 1 } { x ^ { 2 } } d x$
Answer: $\_\_\_\_$

The function $f ( x )$ satisfies $f ( x + 3 ) = f ( x )$ for all real numbers $x$, and $$f ( x ) = \begin{cases} x & ( 0 \leq x < 1 ) \\ 1 & ( 1 \leq x < 2 ) \\ - x + 3 & ( 2 \leq x < 3 ) \end{cases}$$ If $\int _ { - a } ^ { a } f ( x ) d x = 13$, what is the value of the constant $a$? [4 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18

For the quadratic function $f ( x ) = \frac { 3 x - x ^ { 2 } } { 2 }$, a function $g ( x )$ defined on the interval $[ 0 , \infty )$ satisfies the following conditions. (가) When $0 \leq x < 1$, $g ( x ) = f ( x )$. (나) When $n \leq x < n + 1$, $$\begin{aligned} & g ( x ) = \frac { 1 } { 2 ^ { n } } \{ f ( x - n ) - ( x - n ) \} + x \\ & \text{(Here, } n \text{ is a natural number.)} \end{aligned}$$ For some natural number $k ( k \geq 6 )$, the function $h ( x )$ is defined as $$h ( x ) = \begin{cases} g ( x ) & ( 0 \leq x < 5 \text{ or } x \geq k ) \\ 2 x - g ( x ) & ( 5 \leq x < k ) \end{cases}$$ When the sequence $\left\{ a _ { n } \right\}$ is defined by $a _ { n } = \int _ { 0 } ^ { n } h ( x ) d x$ and $\lim _ { n \rightarrow \infty } \left( 2 a _ { n } - n ^ { 2 } \right) = \frac { 241 } { 768 }$, find the value of $k$. [4 points]

An increasing continuous function $f ( x )$ on the set of all real numbers satisfies the following conditions. (가) For all real numbers $x$, $f ( x ) = f ( x - 3 ) + 4$. (나) $\int _ { 0 } ^ { 6 } f ( x ) d x = 0$ What is the area enclosed by the graph of $y = f ( x )$, the $x$-axis, and the two lines $x = 6$ and $x = 9$? [4 points]
(1) 9
(2) 12
(3) 15
(4) 18
(5) 21

Find the value of $\int _ { 1 } ^ { 4 } ( x + | x - 3 | ) d x$. [3 points]

For the function $f ( x ) = \pi \sin 2 \pi x$, a function $g ( x )$ with domain being the set of all real numbers and range being the set $\{ 0,1 \}$, and a natural number $n$ satisfy the following conditions. What is the value of $n$? [4 points]
The function $h ( x ) = f ( n x ) g ( x )$ is continuous on the set of all real numbers and $$\int _ { - 1 } ^ { 1 } h ( x ) d x = 2 , \quad \int _ { - 1 } ^ { 1 } x h ( x ) d x = - \frac { 1 } { 32 }$$
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16

A function $f ( x )$ that is continuous on the set of all real numbers satisfies the following condition. When $n - 1 \leq x < n$, $| f ( x ) | = | 6 ( x - n + 1 ) ( x - n ) |$. (Here, $n$ is a natural number.)
For the function $$g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t - \int _ { x } ^ { 4 } f ( t ) d t$$ defined on the open interval $(0, 4)$, when $g ( x )$ has a minimum value of 0 at $x = 2$, what is the value of $\int _ { \frac { 1 } { 2 } } ^ { 4 } f ( x ) d x$? [4 points]
(1) $- \frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$
(5) $\frac { 5 } { 2 }$

Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function such that $g ( x ) = 1 / x$ if $x \geqslant \mathrm { e } ^ { - 1 }$ and $g ( x ) = 0$ otherwise. We fix a real $\varepsilon \in ] 0 , \mathrm { e } ^ { - 1 } [$. We define two continuous applications $g ^ { + } , g ^ { - } : [ 0,1 ] \rightarrow \mathbb { R }$ as follows:

• $g ^ { + }$ is affine on $\left[ \mathrm { e } ^ { - 1 } - \varepsilon , \mathrm { e } ^ { - 1 } \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } - \varepsilon \right] \cup \left[ \mathrm { e } ^ { - 1 } , 1 \right]$;
• $g ^ { - }$ is affine on $\left[ \mathrm { e } ^ { - 1 } , \mathrm { e } ^ { - 1 } + \varepsilon \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } \left[ \cup \left[ \mathrm { e } ^ { - 1 } + \varepsilon , 1 \right] \right. \right.$.

Calculate $\int _ { 0 } ^ { 1 } g ^ { + } ( t ) \mathrm { d } t$ and $\int _ { 0 } ^ { 1 } g ^ { - } ( t ) \mathrm { d } t$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Let $\varphi \in \mathbb{R}$. Show that, for any complex number $z$ such that $|z| < 1$, $g(z) = \frac{1}{2\pi} \int_{\varphi}^{\varphi + 2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t$.

The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that for all integer $n > 1$,
$$\int _ { 1 } ^ { n } \frac { q ( u ) } { u } \mathrm {~d} u = \ln ( n ! ) + ( n - 1 ) - n \ln ( n ) - \frac { 1 } { 2 } \ln ( n ) = \ln \left( \frac { n ! e ^ { n } } { n ^ { n } \sqrt { n } } \right) - 1$$

Show that for all integer $n > 1$, $$\int_{1}^{n} \frac{q(u)}{u} \mathrm{~d}u = \ln(n!) + (n-1) - n\ln(n) - \frac{1}{2}\ln(n) = \ln\left(\frac{n! e^n}{n^n \sqrt{n}}\right) - 1.$$

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)] $2 - \sqrt{2} + \ln 2$

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

Let $f(x) = \int_0^1 |t - x|\, t\, dt$ for $x \in [0,1]$. Find $f(x)$ and sketch its graph.

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 $[ x ]$ 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 $n$ 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) $\frac { ( n + 2 ) } { 4 }$

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) $\frac { ( n + 2 ) } { 4 }$

6. Let $f ( x ) = x - [ x ]$, for every real number $x$, where $[ x ]$ is the integral part of $x$. Then $\int 1 - 1 f ( x ) d x$ is :
(A) 1
(B) 2
(C) 0
(D) $1 / 2$

8. If for a real number $y , [ y ]$ is the greatest integer less than or equal to $y$, then the value of the integral $\int \sqcap / 23 \sqcap / 2 [ 2 \sin x . d x ]$ is:
(A) $- \pi$
(B) 0
(C) $- \pi / 2$
(D) $\pi / 2$