The function f is such that, for every integer $n$
$$\int _ { n } ^ { n + 1 } \mathrm { f } ( x ) \mathrm { d } x = n + 1$$
Evaluate
$$\sum _ { r = 1 } ^ { 8 } \left( \int _ { 0 } ^ { r } \mathrm { f } ( x ) \mathrm { d } x \right)$$
A 36 B 84 C 120 D 165 E 204 F 288

A sequence of functions $f _ { 1 } , f _ { 2 } , f _ { 3 } , \ldots$ is defined by
$$\begin{aligned} \mathrm { f } _ { 1 } ( x ) & = | x | \\ \mathrm { f } _ { n + 1 } ( x ) & = \left| \mathrm { f } _ { n } ( x ) + x \right| \quad \text { for } n \geq 1 \end{aligned}$$
Find the value of
$$\int _ { - 1 } ^ { 1 } \mathrm { f } _ { 99 } ( x ) \mathrm { d } x$$
A 0
B 0.5
C 1
D 49.5
E 50 F 99 G 99.5 H 100

Given the following statements about a function f

• $\mathrm { f } ^ { \prime \prime } ( x ) = a$ for all $x$
• $\mathrm { f } ( 0 ) = 1 , \mathrm { f } ( 1 ) = 2$
• $\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 1$

find the value of $a$.

Place the following integrals in order of size, starting with the smallest.
$$\begin{aligned} & P = \int _ { 0 } ^ { 1 } 2 ^ { \sqrt { x } } \mathrm {~d} x \\ & Q = \int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x \\ & R = \int _ { 0 } ^ { 1 } ( \sqrt { 2 } ) ^ { x } \mathrm {~d} x \end{aligned}$$
A $P < Q < R$
B $P < R < Q$
C $Q < P < R$
D $Q < R < P$
E $\quad R < P < Q$ F $R < Q < P$

Given that
$$\int _ { 0 } ^ { 1 } ( a x + b ) \mathrm { d } x = 1$$
and
$$\int _ { 0 } ^ { 1 } x ( a x + b ) \mathrm { d } x = 1$$
find the value of $a + b$.

Evaluate
$$\int _ { 9 } ^ { 16 } \left( \frac { 1 } { \sqrt { x } } + \sqrt { x } \right) ^ { 2 } \mathrm {~d} x - \int _ { 9 } ^ { 16 } \left( \frac { 1 } { \sqrt { x } } - \sqrt { x } \right) ^ { 2 } \mathrm {~d} x$$
A 0 B 2 C 4 D 7 E 14 F 28 G 75 H 175

For any integer $n \geq 0$,
$$\int _ { n } ^ { n + 1 } f ( x ) \mathrm { d } x = n + 1$$
Evaluate
$$\int _ { 0 } ^ { 3 } f ( x ) \mathrm { d } x + \int _ { 1 } ^ { 3 } f ( x ) \mathrm { d } x + \int _ { 2 } ^ { 3 } f ( x ) \mathrm { d } x + \int _ { 4 } ^ { 3 } f ( x ) \mathrm { d } x + \int _ { 5 } ^ { 3 } f ( x ) \mathrm { d } x$$

The ceiling of $x$, written $[ x ]$, is defined to be the value of $x$ rounded up to the nearest integer. For example: $\quad \lceil \pi \rceil = 4 , \quad \lceil 2.1 \rceil = 3 , \quad \lceil 8 \rceil = 8$ What is the value of the following integral?
$$\int _ { 0 } ^ { 99 } 2 ^ { \lceil x \rceil } d x$$
A $2 ^ { 99 }$ B $\quad 2 ^ { 99 } - 1$ C $2 ^ { 99 } - 2$ D $2 ^ { 100 }$ E $\quad 2 ^ { 100 } - 1$ F $\quad 2 ^ { 100 } - 2$

Let $f$ be a polynomial with real coefficients. The integral $I _ { p , q }$ where $p < q$ is defined by
$$I _ { p , q } = \int _ { p } ^ { q } ( f ( x ) ) ^ { 2 } - ( f ( | x | ) ) ^ { 2 } \mathrm {~d} x$$
Which of the following statements must be true? $1 I _ { p , q } = 0$ only if $0 < p$ $2 f ^ { \prime } ( x ) < 0$ for all $x$ only if $I _ { p , q } < 0$ for all $p < q < 0$ $3 \quad I _ { p , q } > 0$ only if $p < 0$
A none of them B 1 only C 2 only D 3 only E 1 and 2 only F 1 and 3 only G 2 and 3 only H 1, 2 and 3

$$f''(x) = 6x - 2, \quad f'(0) = 4, \quad f(0) = 1$$
For the function $f$ that satisfies these conditions, what is the value of $f(1)$?
A) 4
B) 5
C) 6
D) 7
E) 8

For the function $f$ whose graph is given above, $$\int_{1}^{3} \frac{x \cdot f'(x) - f(x)}{x^{2}}\, dx$$ What is the value of the integral?
A) $\frac{7}{2}$
B) $\frac{3}{2}$
C) $\frac{2}{3}$
D) $\frac{1}{3}$
E) $\frac{5}{4}$

$$f(x) = \begin{cases} 3 - x, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$$
What is the value of the integral $\displaystyle\int_{1}^{3} f(x+1)\, dx$?
A) 2
B) 4
C) 6
D) 8
E) 10

$$\begin{aligned} & f ^ { \prime } ( x ) = 3 x ^ { 2 } + 4 x + 3 \\ & f ( 0 ) = 2 \end{aligned}$$
Given this, what is the value of $\mathbf { f } ( - \mathbf { 1 } )$?
A) - 2
B) - 1
C) 0
D) 1
E) 2

The slope of the tangent line to the graph of a function f at $\mathrm { x } = \mathrm { a }$ is $1$, and the slope of the tangent line at $x = b$ is $\sqrt { 3 }$. Given that the second derivative function $\mathbf { f } ^ { \prime \prime } ( \mathbf { x } )$ is continuous on the interval $[ \mathbf { a } , \mathbf { b } ]$, what is the value of
$$\int _ { b } ^ { a } f ^ { \prime } ( x ) \cdot f ^ { \prime \prime } ( x ) d x$$
?
A) - 1
B) 1
C) 2
D) $\frac { 1 } { 3 }$
E) $\frac { 2 } { 3 }$

Below, the graph of the derivative of a function f is given. Given that $f ( 0 ) = 1$, what is the value of $f ( 2 )$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { - 1 } { 2 }$
E) $\frac { - 1 } { 3 }$

For a continuous function f defined on the set of real numbers,
$$\int _ { 1 } ^ { 3 } f ( x ) d x = 5$$
is known. Accordingly,
$$\int _ { 0 } ^ { 1 } ( 4 + f ( 2 x + 1 ) ) d x$$
What is the value of this integral?
A) 1
B) 2
C) 3
D) $\frac { 5 } { 2 }$
E) $\frac { 13 } { 2 }$

Let $n$ be a natural number,
$$\begin{aligned} & f _ { n } : [ n , n + 1 ) \rightarrow \left[ 0 , \frac { 1 } { 2 ^ { n } } \right) \\ & f _ { n } ( x ) = \frac { ( x - n ) ^ { 2 } } { 2 ^ { n } } \end{aligned}$$
The regions between the functions defined in this form and the x-axis are given shaded in the figure below.
Accordingly, what is the sum of the areas of all shaded regions in square units?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 8 } { 9 }$
E) $\frac { 11 } { 12 }$

The derivative of a function f that is defined and differentiable on the set of real numbers is given as
$$f ^ { \prime } ( x ) = \begin{cases} 1 , & \text{if } x \leq 1 \\ x , & \text{if } x > 1 \end{cases}$$
Given that $f ( 1 ) = 1$, what is the value of $f ( 0 ) + f ( 3 )$?
A) 2
B) 3
C) 4
D) 5
E) 6

For every integer $m$ greater than 1
$$\int \tan ^ { m } x d x = \frac { 1 } { m - 1 } \tan ^ { m - 1 } x - \int \tan ^ { m - 2 } x d x$$
the equality is satisfied. Accordingly, what is the value of the integral $\int _ { 0 } ^ { \frac { \pi } { 4 } } \tan ^ { 4 } \mathrm { xdx }$?
A) $\frac { 2 \pi + 3 } { 4 }$
B) $\frac { 4 \pi - 3 } { 8 }$
C) $\frac { 3 \pi - 8 } { 12 }$
D) $\pi + 2$
E) $2 \pi + 1$

$f ( x ) = \begin{cases} 2 x - 4 , & \text{if } 0 \leq x < 1 \\ - 2 , & \text{if } 1 \leq x < 4 \\ x - 6 , & \text{if } 4 \leq x \leq 6 \end{cases}$
Given this, what is the value of the integral $\int _ { 0 } ^ { 6 } f ( x ) d x$?
A) - 11
B) - 10
C) - 9
D) - 8
E) - 7

Let a be a positive real number. For every second-degree polynomial $P ( x )$ with real coefficients and leading coefficient 1,
$$\int _ { - 1 } ^ { 1 } \mathrm { P } ( \mathrm { x } ) \mathrm { dx } = \mathrm { P } ( \mathrm { a } ) + \mathrm { P } ( - \mathrm { a } )$$
the equality is satisfied. Accordingly, what is the value of a?
A) $\sqrt { 2 }$
B) $\sqrt { 3 }$
C) $\sqrt { 6 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E) $\frac { \sqrt { 3 } } { 3 }$

$\int_{\pi/6}^{?} 2 \tan ( 2 x ) \, d x$\ What is the value of the integral?\ A) $\ln 2$\ B) $\ln 3$\ C) $\ln 4$\ D) $\ln 5$\ E) $\ln 6$

For an increasing and continuous function f defined on the set of real numbers,
$$\begin{aligned} & f ( 0 ) = 2 \\ & f ( 1 ) = 3 \\ & f ( 2 ) = 4 \end{aligned}$$
equalities are given.
Accordingly, the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ could be which of the following?
A) 4 B) 4.5 C) 6 D) 7.5 E) 8

In the rectangular coordinate plane, the graph of $f ^ { \prime }$, the derivative of function $f$, is given on the closed interval $[ 0,10 ]$. The areas of the regions between this graph and the x-axis are shown as follows.
$$f ( 0 ) = \frac { - 1 } { 2 }$$
Given that, how many different roots does the function $f$ have on the interval $[ 0 , 10 ]$?
A) 1
B) 2
C) 3
D) 4
E) 5

Let $a$ and $b$ be real numbers. A function $f$ that is continuous on the set of real numbers is defined as
$$f ( x ) = \begin{cases} 6 - \frac { 3 x ^ { 2 } } { 2 } , & x < 2 \\ a x - b & x \geq 2 \end{cases}$$
$$\int _ { 0 } ^ { 4 } f ( x ) d x = \int _ { 2 } ^ { 6 } f ( x ) d x$$
Given that, what is the sum $a + b$?
A) 1
B) 2
C) 3
D) 4
E) 5