The integral $80\int_0^{\frac{\pi}{4}} \left(\frac{\sin\theta + \cos\theta}{9 + 16\sin 2\theta}\right)d\theta$ is equal to:
(1) $3\log_e 4$
(2) $4\log_e 3$
(3) $6\log_e 4$
(4) $2\log_e 3$

If $f ( x ) = \int \frac { 1 } { x ^ { 1 / 4 } \left( 1 + x ^ { 1 / 4 } \right) } \mathrm { d } x , f ( 0 ) = - 6$, then $f ( 1 )$ is equal to :
(1) $4 \left( \log _ { e } 2 - 2 \right)$
(2) $2 - \log _ { e ^ { 2 } } 2$
(3) $\log _ { e } 2 + 2$
(4) $4 \left( \log _ { e } 2 + 2 \right)$

Let $\mathrm { I } ( x ) = \int \frac { d x } { ( x - 11 ) ^ { \frac { 11 } { 13 } } ( x + 15 ) ^ { \frac { 15 } { 13 } } }$. If $\mathrm { I } ( 37 ) - \mathrm { I } ( 24 ) = \frac { 1 } { 4 } \left( \frac { 1 } { \mathrm {~b} ^ { \frac { 1 } { 13 } } } - \frac { 1 } { \mathrm { c } ^ { \frac { 1 } { 13 } } } \right) , \mathrm { b } , \mathrm { c } \in \mathrm { N }$, then $3 ( \mathrm {~b} + \mathrm { c } )$ is equal to
(1) 22
(2) 39
(3) 40
(4) 26

If $\int \frac{2x^{2} + 5x + 9}{\sqrt{x^{2} + x + 1}} \, \mathrm{d}x = x\sqrt{x^{2} + x + 1} + \alpha\sqrt{x^{2} + x + 1} + \beta \log_{e}\left| x + \frac{1}{2} + \sqrt{x^{2} + x + 1} \right| + \mathrm{C}$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to $\_\_\_\_$.

Let $f ( x ) = 4 \sqrt { 3 } e ^ { - x } \cos x + 6 e ^ { - x }$.
(1) Let $a$ and $b$ ($a < b$) be the values of $x$ satisfying $f ( x ) = 0$ on $0 \leqq x < 2 \pi$. Then,
$$a = \frac { \mathbf{A} } { \mathbf{B} } \pi , \quad b = \frac { \mathbf{C} } { \mathbf{D} } \pi$$
(2) The values of the constants $p$ and $q$ satisfying
$$\frac { d } { d x } \left( p e ^ { - x } \cos x + q e ^ { - x } \sin x \right) = e ^ { - x } \cos x$$
are given by
$$p = \frac { \mathbf { E F } } { \mathbf { G } } , \quad q = \frac { \mathbf { H } } { \mathbf { I } } .$$
(3) Using the values of $a$ and $b$ obtained in (1), we set $A = e ^ { - a }$ and $B = e ^ { - b }$. When we calculate the value of $\int _ { a } ^ { b } f ( x ) d x$, we obtain
$$\int _ { a } ^ { b } f ( x ) d x = ( \mathbf { J } - \sqrt { \mathbf{J} } \mathbf { K } ) A - ( \mathbf { L } + \sqrt { \mathbf{L} } ) B .$$

Consider the definite integral $S = \int _ { 0 } ^ { a } x \sqrt { \frac { 1 } { 3 } x + 2 } \, d x$.
(1) Set $t = \sqrt { \frac { 1 } { 3 } x + 2 }$. Then we have
$$\begin{aligned} \int x \sqrt { \frac { 1 } { 3 } x + 2 } \, d x & = \mathbf { N O } \int \left( t ^ { \mathbf { P } } - \mathbf { Q } t ^ { \mathbf { R } } \right) d t \\ & = \mathbf { S } + C , \end{aligned}$$
where $C$ is the integral constant.
(2) Using the result in (1), we have
$$S = \mathbf { T } .$$
Thus we obtain
$$\lim _ { a \rightarrow \infty } \frac { S } { a ^ { \frac { \mathbf { U } } { \mathbf{V} } } } = \frac { \mathbf { W } \sqrt { \mathbf { X } } } { \mathbf { Y Z } }$$
For $\mathbf{S}$ and $\mathbf{T}$, choose the appropriate expression from among the choices (0) $\sim$ (9) below.
(0) $\frac { 6 } { 5 } t ^ { 5 } \left( 3 t ^ { 2 } - 10 \right)$
(1) $\frac { 6 } { 5 } t ^ { 3 } \left( 3 t ^ { 2 } - 10 \right)$
(2) $\frac { 12 } { 5 } t ^ { 5 } \left( 3 t ^ { 2 } - 5 \right)$
(3) $\frac { 12 } { 5 } t ^ { 3 } \left( 3 t ^ { 2 } - 5 \right)$
(4) $\frac { 6 } { 5 } t ^ { 3 } \left( 3 t ^ { 2 } - 5 \right)$
(5) $\frac { 6 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 5 } ( a - 4 ) + 8 \sqrt { 2 } \right\}$
(6) $\frac { 12 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 3 } ( a - 2 ) + 4 \sqrt { 2 } \right\}$
(7) $\frac { 12 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 5 } ( a - 2 ) + 4 \sqrt { 2 } \right\}$
(8) $\frac { 6 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 3 } ( a - 4 ) + 8 \sqrt { 2 } \right\}$
(9) $\frac { 6 } { 5 } \left\{ \left( \sqrt { \frac { 1 } { 3 } a + 2 } \right) ^ { 3 } ( a - 2 ) + 8 \sqrt { 2 } \right\}$

Consider the function $$f(x) = \frac{\sin x}{3 - 2\cos x} \quad (0 \leqq x \leqq \pi)$$
(1) The derivative of $f(x)$ is $$f'(x) = \frac{\mathbf{A}\cos x - \mathbf{B}}{(\mathbf{C} - \mathbf{D}\cos x)^2}.$$ Let $\alpha$ be the value of $x$ at which $f(x)$ has a local extremum. Then we have $$\cos\alpha = \frac{\mathbf{E}}{\mathbf{F}}.$$
(2) The portion of the plane bounded by the graph of the function $y = f(x)$ and the $x$-axis is divided into two parts by the straight line $x = \alpha$. Let $S_1$ be the area of the part located on the left side of the line. Then we have $$S_1 = \int_{\frac{\mathbf{G}}{\mathbf{H}}}^{\mathbf{I}} \frac{dt}{\mathbf{J} - \mathbf{K}t} = \frac{\mathbf{L}}{\mathbf{L}}\log\frac{\mathbf{L}}{\mathbf{L}}.$$ Let $S_2$ be the area of the part located on the right side. We have $$S_2 = \frac{\mathbf{P}}{2}\log\mathbf{Q}.$$

Calculate the following definite integral: $$I = \int_{0}^{\sin\theta} \frac{\arctan(\arcsin x)}{\sqrt{1 - x^{2}}} \mathrm{~d}x$$ where $0 < \theta < \pi/2$.

$$\int_{0}^{4} \frac{6x}{\sqrt{2x+1}}\, dx$$
What is the value of the integral?
A) 12
B) 15
C) 18
D) 20
E) 24

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) $\frac { 2 } { 3 }$

In the integral $\int \frac { \ln \sqrt { x } } { \sqrt { x } } d x$, if the substitution $u = \sqrt { x }$ is made, which of the following integrals is obtained?
A) $\int \ln u \, d u$
B) $\int 2 \ln u \, d u$
C) $\int \frac { \ln u } { u } d u$
D) $\int \frac { \ln u } { 2 u } d u$
E) $\int u \ln u \, d u$

$$\int _ { 4 } ^ { 9 } \frac { \sqrt { x } } { x - 1 } d x$$
If the substitution $\mathbf { u } = \sqrt { \mathbf { x } }$ is made in the integral, which of the following integrals is obtained?
A) $\int _ { 4 } ^ { 9 } \frac { u } { u ^ { 2 } - 1 } d u$
B) (missing option)
C) $\int _ { 2 } ^ { 3 } \frac { u } { 2 \left( u ^ { 2 } - 1 \right) } d u$
D) $\int _ { 2 } ^ { 3 } \frac { 2 u ^ { 2 } } { u ^ { 2 } - 1 } d u$
E) $\int _ { 2 } ^ { 3 } \frac { u } { u ^ { 2 } - 1 } d u$

$$\int _ { 4 } ^ { 9 } \frac { 3 x - 3 } { \sqrt { x } + 1 } d x$$
What is the value of the integral?
A) 13
B) 18
C) 23
D) 28
E) 33

$\int \sqrt { 1 + e^{x^{2}} } \, d x$\ In the integral, if the substitution $u = \sqrt { 1 + e ^ { x } }$ is made, which of the following integrals is obtained?\ A) $\int \frac { 2 u } { u ^ { 2 } + 1 } d u$\ B) $\int \frac { u ^ { 2 } } { u ^ { 2 } + 1 } d u$\ C) $\int \frac { 1 } { u ^ { 2 } - 1 } d u$\ D) $\int \frac { u } { u ^ { 2 } - 1 } d u$\ E) $\int \frac { 2 u ^ { 2 } } { u ^ { 2 } - 1 } d u$

$$\int \frac { ( 3 \sqrt { x } + 2 ) ^ { 5 } } { \sqrt { x } } d x$$
Which of the following is this integral equal to? (c is an arbitrary constant.)
A) $\frac { 1 } { 18 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
B) $\frac { 1 } { 9 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
C) $\frac { 2 } { 9 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
D) $\frac { 1 } { 3 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
E) $\frac { 2 } { 3 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$

$$\int_{1}^{2} (x+2) \cdot \sqrt[3]{x^{2} + 4x - 4}\, dx$$
What is the value of this integral?
A) $\dfrac{45}{8}$ B) $\dfrac{47}{8}$ C) $\dfrac{49}{8}$ D) $\dfrac{45}{4}$ E) $\dfrac{47}{4}$