Consider the function $f(x) = x^{\cos(x) + \sin(x)}$ defined for $x \geq 0$.
(a) Prove that
$$0.4 \leq \int_{0}^{1} f(x)\, dx \leq 0.5$$
(b) Suppose the graph of $f(x)$ is being traced on a computer screen with the uniform speed of 1 cm per second (i.e., this is how fast the length of the curve is increasing). Show that at the moment the point corresponding to $x = 1$ is being drawn, the $x$ coordinate is increasing at the rate of
$$\frac{1}{\sqrt{2 + \sin(2)}} \text{ cm per second.}$$

Write the values of the following.
(a) $\int_{-3}^{3} \left| 3x^{2} - 3 \right| dx$.
(b) $f'(1)$ where $f(t) = \int_{0}^{t} \left| 3x^{2} - 3 \right| dx$.

Compute the following integral $$\int_{0}^{\frac{\pi}{2}} \frac{\mathrm{~d}x}{(\sqrt{\sin x} + \sqrt{\cos x})^{4}}.$$

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Determine (with appropriate justification) the following limit: $$\lim_{n \longrightarrow \infty} \int_0^1 nx^n f(x)\,\mathrm{d}x$$

Calculate the following two definite integrals. It may be useful to first sketch the graph. $$\int_{1}^{e^{2}} \ln|x|\, dx \qquad \int_{-1}^{1} \frac{\ln|x|}{|x|}\, dx$$

(a) Find the domain of the function $g(x)$ defined by the following formula. $$g(x) = \int_{10}^{x} \log_{10}\left(\log_{10}\left(t^2 - 1000t + 10^{1000}\right)\right) dt$$ Calculate the quantities below. You may give an approximate answer where necessary, but clearly state which answers are exact and which are approximations.
(b) $g(1000)$.
(c) $x$ in $[10, 1000]$ where $g(x)$ has the maximum possible slope.
(d) $x$ in $[10, 1000]$ where $g(x)$ has the least possible slope.
(e) $\lim_{x \rightarrow \infty} \frac{\ln(x)}{g(x)}$ if it exists.

The following is a graph showing the velocity $v ( t )$ at time $t$ ( $0 \leqq t \leqq d$ ) of a point P moving on a number line starting from the origin.
When $\int _ { 0 } ^ { a } | v ( t ) | d t = \int _ { a } ^ { d } | v ( t ) | d t$, which of the following statements in are correct? (Here, $0 < a < b < c < d$.) [3 points]
Remarks ㄱ. Point P passes through the origin again after starting. ㄴ. $\int _ { 0 } ^ { c } v ( t ) d t = \int _ { c } ^ { d } v ( t ) d t$ ㄷ. $\int _ { 0 } ^ { b } v ( t ) d t = \int _ { b } ^ { d } | v ( t ) | d t$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

A polynomial function $f ( x )$ satisfies $$\int _ { 1 } ^ { x } f ( t ) d t = x ^ { 3 } - 2 a x ^ { 2 } + a x$$ for all real numbers $x$. Find the value of $f ( 3 )$. (Here, $a$ is a constant.) [3 points]

For the function $f ( x ) = x ^ { 3 } + x$, find the value of $\lim _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } f \left( 1 + \frac { 2 k } { n } \right)$. [3 points]

For a natural number $n \geq 2$, let $C _ { n }$ be the circle obtained by translating the circle $C$ with center at the origin and radius 1 by $\frac { 2 } { n }$ in the $x$-direction. Let $l _ { n }$ be the length of the common chord of circles $C$ and $C _ { n }$. When $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { \left( n l _ { n } \right) ^ { 2 } } = \frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

When the function $f ( x ) = 6 x ^ { 2 } + 2 a x$ satisfies $\int _ { 0 } ^ { 1 } f ( x ) d x = f ( 1 )$, what is the value of the constant $a$? [2 points]
(1) $- 4$
(2) $- 2$
(3) 0
(4) 2
(5) 4

(Calculus) A continuous function $f(x)$ defined on the closed interval $[0, 1]$ satisfies $f(0) = 0$, $f(1) = 1$, has a second derivative on the open interval $(0, 1)$, and $f'(x) > 0$, $f''(x) > 0$. Which of the following is equal to $\int_0^1 \{f^{-1}(x) - f(x)\} dx$? [3 points]
(1) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{2n}$
(2) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{2}{n}$
(3) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
(4) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{2n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
(5) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{2k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$

There is a function $f ( x ) = x ^ { 2 } + a x + b \quad ( a \geqq 0 , b > 0 )$. For a natural number $n \geq 2$, divide the closed interval $[ 0,1 ]$ into $n$ equal parts, and let the division points (including both endpoints) be $$0 = x _ { 0 } , x _ { 1 } , x _ { 2 } , \cdots , x _ { n - 1 } , x _ { n } = 1$$ respectively. Let $A _ { k }$ be the area of the rectangle with base $\left[ x _ { k - 1 } , x _ { k } \right]$ and height $f \left( x _ { k } \right)$. $( k = 1,2 , \cdots , n )$
Given that the sum of the areas of the two rectangles at the ends is $$A _ { 1 } + A _ { n } = \frac { 7 n ^ { 2 } + 1 } { n ^ { 3 } }$$ find the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 8 k } { n } A _ { k }$. [4 points]

There is a function $f ( x )$ that is differentiable on the set of all real numbers. For all real numbers $x$, $f ( 2 x ) = 2 f ( x ) f ^ { \prime } ( x )$, and $$f ( a ) = 0 , \quad \int _ { 2 a } ^ { 4 a } \frac { f ( x ) } { x } d x = k \quad ( a > 0,0 < k < 1 )$$ When this holds, what is the value of $\int _ { a } ^ { 2 a } \frac { \{ f ( x ) \} ^ { 2 } } { x ^ { 2 } } d x$ expressed in terms of $k$? [3 points]
(1) $\frac { k ^ { 2 } } { 4 }$
(2) $\frac { k ^ { 2 } } { 2 }$
(3) $k ^ { 2 }$
(4) $k$
(5) $2 k$

A quadratic function $f ( x )$ satisfies $f ( 0 ) = - 1$ and
$$\int _ { - 1 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { - 1 } ^ { 0 } f ( x ) d x$$
What is the value of $f ( 2 )$? [4 points]
(1) 11
(2) 10
(3) 9
(4) 8
(5) 7

Find the value of $\int _ { 0 } ^ { 5 } ( 4 x - 3 ) d x$. [3 points]

For the function $f(x) = x + 1$, $$\int _ { -1 } ^ { 1 } \{ f(x) \} ^ { 2 } dx = k \left( \int _ { -1 } ^ { 1 } f(x) dx \right) ^ { 2 }$$ what is the value of the constant $k$? [3 points]
(1) $\frac{1}{6}$
(2) $\frac{1}{3}$
(3) $\frac{1}{2}$
(4) $\frac{2}{3}$
(5) $\frac{5}{6}$

A continuous function $f ( x )$ satisfies
$$f ( x ) = e ^ { x ^ { 2 } } + \int _ { 0 } ^ { 1 } t f ( t ) d t$$
What is the value of $\int _ { 0 } ^ { 1 } x f ( x ) d x$? [3 points]
(1) $e - 2$
(2) $\frac { e - 1 } { 2 }$
(3) $\frac { e } { 2 }$
(4) $e - 1$
(5) $\frac { e + 1 } { 2 }$

For the cubic function $f(x) = x^3 - 3x + a$, the function $$F(x) = \int_{0}^{x} f(t)\, dt$$ has exactly one extremum. What is the minimum value of the positive number $a$? [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The graph of a continuous function $y = f ( x )$ is symmetric about the origin, and for all real numbers $x$, $$f ( x ) = \frac { \pi } { 2 } \int _ { 1 } ^ { x + 1 } f ( t ) d t$$ When $f ( 1 ) = 1$, what is the value of $$\pi ^ { 2 } \int _ { 0 } ^ { 1 } x f ( x + 1 ) d x$$ ? [4 points]
(1) $2 ( \pi - 2 )$
(2) $2 \pi - 3$
(3) $2 ( \pi - 1 )$
(4) $2 \pi - 1$
(5) $2 \pi$

For a real number $a$, when $\int _ { - a } ^ { a } \left( 3 x ^ { 2 } + 2 x \right) d x = \frac { 1 } { 4 }$, find the value of $50 a$. [3 points]

What is the value of $\int _ { 0 } ^ { 1 } 3 \sqrt { x } \, d x$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

If $\int _ { 0 } ^ { 1 } ( 2 x + a ) d x = 4$, what is the value of the constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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

What is the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } 2 \sin x \, d x$? [2 points]
(1) 0
(2) $\frac { 1 } { 2 }$
(3) 1
(4) $\frac { 3 } { 2 }$
(5) 2