Consider the function $$f(x) = \sin x + \frac{\sin 2x}{2} + \frac{\sin 3x}{3}$$ on the interval $0 \leqq x \leqq \pi$. We are to show that $f(x) > 0$ on $0 < x < \pi$, and to find the area $S$ of the region bounded by the graph of $y = f(x)$ and the $x$-axis.
(1) For $\mathbf{K}$, $\mathbf{N}$, $\mathbf{Q}$, $\mathbf{R}$ in the following sentences, choose the correct answer from the following two choices: (0) increasing, (1) decreasing, and for the other blanks, enter the correct number.
When we differentiate $f(x)$, we have $$f'(x) = (\mathbf{A}\cos^2 x - \mathbf{B})(\mathbf{C}\cos x + \mathbf{D}).$$ Hence, over the range $0 \leqq x \leqq \pi$, there are three $x$'s at which $f'(x) = 0$, and when they are arranged in ascending order, they are given accordingly.
Next, looking at whether $f(x)$ is increasing or decreasing, the behaviour is described accordingly.
Also, we have $$f(0) = 0, \quad f(\pi) = 0, \quad f\left(\frac{\mathbf{L}}{\mathbf{M}}\pi\right) = \frac{\sqrt{\mathbf{S}}}{\mathbf{T}} > 0.$$ Hence we see that $f(x) > 0$ on $0 < x < \pi$.
(2) The area $S$ of the region bounded by the graph of $y = f(x)$ and the $x$-axis is $$S = \frac{\mathbf{UV}}{\mathbf{W}}.$$

Let the function $f ( x ) = \left\{ \begin{array} { l l l } x ^ { 3 } e ^ { - 1 / x ^ { 2 } } & \text { if } & x \neq 0 \\ 0 & \text { if } & x = 0 \end{array} \right.$ a) (1 point) Study the continuity and differentiability of $f ( x )$ at $x = 0$. b) ( 0.5 points) Study whether $f ( x )$ presents any type of even or odd symmetry. c) (1 point) Calculate the following integral: $\int _ { 1 } ^ { 2 } \frac { f ( x ) } { x ^ { 6 } } d x$

Let the function
$$f ( x ) = \begin{cases} x & \text { if } x \leq 0 \\ x \ln ( x ) & \text { if } x > 0 \end{cases}$$
a) ( 0.5 points) Study the continuity and differentiability of $f ( x )$ at $x = 0$. b) (1 point) Study the intervals of increase and decrease of $\mathrm { f } ( \mathrm { x } )$, as well as the relative maxima and minima. c) (1 point) Calculate $\int _ { 1 } ^ { 2 } f ( x ) d x$.

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$. Given that the slope of the tangent line at the inflection point of the graph of $y = f ( x )$ is 4, so that $f ^ { \prime } ( x )$ is as found in question 14.
Find the value of $\int _ { - 3 } ^ { 1 } f ^ { \prime } ( x ) \, d x$.

Find the value of
$$\int _ { 1 } ^ { 2 } \left( x ^ { 2 } - \frac { 4 } { x ^ { 2 } } \right) ^ { 2 } d x$$

The two functions $F ( n )$ and $G ( n )$ are defined as follows for positive integers $n$ :
$$\begin{aligned} & F ( n ) = \frac { 1 } { n } \int _ { 0 } ^ { n } ( n - x ) d x \\ & G ( n ) = \sum _ { r = 1 } ^ { n } F ( r ) \end{aligned}$$
What is the smallest positive integer $n$ such that $G ( n ) > 150$ ?
A 22
B 23
C 24
D 25
E 26

Given that
$$2\int_0^1 f(x) \, dx + 5\int_1^2 f(x) \, dx = 14$$
and
$$\int_0^1 f(x+1) \, dx = 6$$
find the value of
$$\int_0^2 f(x) \, dx$$

The curve $y = x ^ { 3 } - 6 x + 3$ has turning points at $x = \alpha$ and $x = \beta$, where $\beta > \alpha$. Find
$$\int _ { \alpha } ^ { \beta } x ^ { 3 } - 6 x + 3 \mathrm {~d} x$$
A $- 8 \sqrt { 2 }$ B - 10 C $- 10 + 6 \sqrt { 2 }$ D 0 E $\quad 12 - 8 \sqrt { 2 }$ F $\quad 6 \sqrt { 2 }$ G 12

The function f is such that $\mathrm { f } ( 0 ) = 0$, and $x \mathrm { f } ( x ) > 0$ for all non-zero values of $x$. It is given that
$$\int _ { - 2 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 4$$
and
$$\int _ { - 2 } ^ { 2 } | \mathrm { f } ( x ) | \mathrm { d } x = 8$$
Evaluate
$$\int _ { - 2 } ^ { 0 } \mathrm { f } ( | x | ) \mathrm { d } x$$
A - 8 B - 6 C - 4 D - 2 E 2 F $\quad 4$ G 6 H 8

Find the value of
$$\int _ { 1 } ^ { 4 } \left( 3 \sqrt { x } + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x$$
A - 0.75
B 7.125
C 11
D 17
E 18 F 21.875 G 34.5

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 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 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) $\frac { \sqrt { 3 } } { 3 }$

The graph of a function f defined on the interval [-1, 7] is given in the rectangular coordinate plane divided into unit squares as shown in the figure.
Accordingly, what is the value of the integral $\int _ { - 1 } ^ { 7 } f ( x ) d x$?
A) 2
B) 4
C) 6
D) 8
E) 10

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

The volume of a right circular cylinder with radius $r$ and height $h$ is calculated using the formula $\mathrm{V} = \pi r^{2} \mathrm{~h}$.
Two right circular cylinders with equal heights, empty interiors, and parallel bases are nested inside each other, with two faucets on top. One of these faucets fills the inner cylinder, while the other fills the region between the cylinders, with the same amount of water per unit time.
The faucets are opened simultaneously and closed when the inner cylinder is completely filled. In the final state, the height of the water in the inner cylinder is 4 times the height of the water in the region between the cylinders.
Accordingly, what is the ratio of the radius of the outer cylinder to the radius of the inner cylinder?
A) $\sqrt{3}$
B) $\sqrt{5}$
C) $\sqrt{7}$
D) 2
E) 3