Let $y = f(x)$ be a thrice differentiable function on $(-5, 5)$. Let the tangents to the curve $y = f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x$-axis. If $27\int_1^3 \left(f'(t)\right)^2 + 1\right) f''(t)\, dt = \alpha + \beta\sqrt{3}$ where $\alpha, \beta$ are integers, then the value of $\alpha + \beta$ equals
(1) $-14$
(2) 26
(3) $-16$
(4) 36

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

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) $2\log_e 3$

Q76. The integral $\int _ { 0 } ^ { \pi / 4 } \frac { 136 \sin x } { 3 \sin x + 5 \cos x } d x$ is equal to :
(1) $3 \pi - 50 \log _ { e } 2 + 20 \log _ { e } 5$
(2) $3 \pi - 25 \log _ { e } 2 + 10 \log _ { e } 5$
(3) $3 \pi - 10 \log _ { e } ( 2 \sqrt { 2 } ) + 10 \log _ { e } 5$
(4) $3 \pi - 30 \log _ { e } 2 + 20 \log _ { e } 5$

Q75. $\int _ { 0 } ^ { \pi / 4 } \frac { \cos ^ { 2 } x \sin ^ { 2 } x } { \left( \cos ^ { 3 } x + \sin ^ { 3 } x \right) ^ { 2 } } d x$ is equal to
(1) $1 / 6$
(2) $1 / 3$
(3) $1 / 12$
(4) $1 / 9$

Q88. If $\int \frac { 1 } { \sqrt [ 5 ] { ( x - 1 ) ^ { 4 } ( x + 3 ) ^ { 6 } } } \mathrm {~d} x = \mathrm { A } \left( \frac { \alpha x - 1 } { \beta x + 3 } \right) ^ { B } + \mathrm { C }$, where C is the constant of integration, then the value of $\alpha + \beta + 20 \mathrm { AB }$ is $\_\_\_\_$

Q73. Let $\int \frac { 2 - \tan x } { 3 + \tan x } \mathrm {~d} x = \frac { 1 } { 2 } \left( \alpha x + \log _ { \mathrm { e } } | \beta \sin x + \gamma \cos x | \right) + C$, where $C$ is the constant of integration. Then $\alpha + \frac { \gamma } { \beta }$ is equal to :
(1) 7
(2) 4
(3) 1
(4) 3

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
$$\int _ { a } ^ { b } f ( x ) d x = ( \mathbf { J } - \sqrt { \mathbf{J} } \mathbf { K } ) A - ( \mathbf { L } + \sqrt { \mathbf{L} } ) B .$$

Given the function $f ( x ) = \frac { x ^ { 2 } + x + 6 } { x - 2 }$, it is requested:
a) ( 0.5 points) Determine its domain and vertical asymptotes.
b) ( 0.5 points) Calculate $\lim _ { x \rightarrow \infty } \frac { f ( x ) } { x }$.
c) (1 point) Calculate $\int _ { 3 } ^ { 5 } f ( x ) d x$.

a) (1.25 points) Calculate, if they exist, the value of the following limits:
a.1) $(0.5$ points) $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } ( 1 - 2 x ) } { x - 2 x ^ { 2 } - \operatorname { sen } x }$
a.2) (0.75 points) $\lim _ { x \rightarrow \infty } \frac { 1 } { x } \left( \frac { 3 } { x } - \frac { 2 } { \operatorname { sen } \frac { 1 } { x } } \right)$
(Hint: use the change of variable $t = 1 / x$ where necessary).
b) (1.25 points) Calculate the following integrals:
b.1) (0.5 points) $\int \frac { x } { x ^ { 2 } - 1 } d x$
b.2) (0.75 points) $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x } d x$

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) $\frac { 2 } { 3 }$

$$\int _ { 2 } ^ { 3 } \frac { 2 x ^ { 2 } } { x ^ { 2 } - 1 } d x$$
What is the value of the integral?
A) $1 + \ln \left( \frac { 4 } { 3 } \right)$
B) $1 + \ln \left( \frac { 9 } { 2 } \right)$
C) $2 + \ln \left( \frac { 3 } { 2 } \right)$
D) $2 + \ln \left( \frac { 5 } { 3 } \right)$
E) $3 + \ln \left( \frac { 1 } { 3 } \right)$

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

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