The function $f ( x ) = 3 x ^ { 2 } - a x$ satisfies
$$\lim _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } f \left( \frac { 3 k } { n } \right) = f ( 1 )$$
Find the value of the constant $a$. [4 points]

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

What is the value of $\int _ { e } ^ { e ^ { 2 } } \frac { \ln x - 1 } { x ^ { 2 } } d x$? [3 points]
(1) $\frac { e - 2 } { e ^ { 2 } }$
(2) $\frac { e - 1 } { e ^ { 2 } }$
(3) $\frac { 1 } { e }$
(4) $\frac { e + 1 } { e ^ { 2 } }$
(5) $\frac { e + 2 } { e ^ { 2 } }$

For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ Using the change of variable $x = 2\sin t$, calculate $m_{0}$.

For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ Using the change of variable $x = 2\sin t$, calculate $m_{0}$.

Deduce the values of $$\int _ { 0 } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } } \mathrm {~d} u \quad \text { then of } \quad \int _ { - \infty } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } / 2 } \mathrm {~d} u .$$

If $b = \int _ { 0 } ^ { 1 } \frac { e ^ { t } } { t + 1 } d t$ then $\int _ { a - 1 } ^ { a } \frac { e ^ { - t } } { t - a - 1 } d t$ is
(A) $b e ^ { a }$
(B) $b e ^ { - a }$
(C) $- b e ^ { - a }$
(D) $- b e ^ { a }$

If $b = \int _ { 0 } ^ { 1 } \frac { e ^ { t } } { t + 1 } d t$ then $\int _ { a - 1 } ^ { a } \frac { e ^ { - t } } { t - a - 1 } d t$ is
(A) $b e ^ { a }$
(B) $b e ^ { - a }$
(C) $- b e ^ { - a }$
(D) $- b e ^ { a }$

If $b = \int _ { 0 } ^ { 1 } \frac { e ^ { t } } { t + 1 } d t$ then $\int _ { a - 1 } ^ { a } \frac { e ^ { - t } } { t - a - 1 } d t$ is
(A) $b e ^ { a }$
(B) $b e ^ { - a }$
(C) $- b e ^ { - a }$
(D) $- b e ^ { a }$

The value of the integral
$$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 + \sqrt { 3 } } { \left( ( x + 1 ) ^ { 2 } ( 1 - x ) ^ { 6 } \right) ^ { \frac { 1 } { 4 } } } d x$$
is $\_\_\_\_$ .

The integral $\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { \ln ( 1 + 2 x ) } { 1 + 4 x ^ { 2 } } d x$ equals
(1) $\frac { \pi } { 4 } \ln 2$
(2) $\frac { \pi } { 16 } \ln 2$
(3) $\frac { \pi } { 8 } \ln 2$
(4) $\frac { \pi } { 32 } \ln 2$

The integral $\int_0^{\pi/4} \frac{\sin x + \cos x}{9 + 16\sin 2x} dx$ is equal to:
(1) $\frac{1}{20} \log 3$
(2) $\log 3$
(3) $\frac{1}{20} \log 9$
(4) $\frac{1}{10} \log 3$

If $\int_0^{\pi/3} \frac{\tan\theta}{\sqrt{2k\sec\theta}}\,d\theta = 1 - \frac{1}{\sqrt{2}},\,(k > 0)$, then the value of $k$ is
(1) $\frac{1}{2}$
(2) 1
(3) 2
(4) 4

The value of the integral $\int _ { 0 } ^ { 1 } \frac { \sqrt { x } d x } { ( 1 + x ) ( 1 + 3 x ) ( 3 + x ) }$ is: (1) $\frac { \pi } { 4 } \left( 1 - \frac { \sqrt { 3 } } { 2 } \right)$ (2) $\frac { \pi } { 8 } \left( 1 - \frac { \sqrt { 3 } } { 6 } \right)$ (3) $\frac { \pi } { 8 } \left( 1 - \frac { \sqrt { 3 } } { 2 } \right)$ (4) $\frac { \pi } { 4 } \left( 1 - \frac { \sqrt { 3 } } { 6 } \right)$

The integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { 1 } { 3 + 2 \sin x + \cos x } d x$ is equal to:
(1) $\tan ^ { - 1 } ( 2 )$
(2) $\tan ^ { - 1 } ( 2 ) - \frac { \pi } { 4 }$
(3) $\frac { 1 } { 2 } \tan ^ { - 1 } ( 2 ) - \frac { \pi } { 8 }$
(4) $\frac { 1 } { 2 }$

Let $[ x ]$ denote the greatest integer function and $f ( x ) = \max \{ 1 + x + [ x ] , 2 + x , x + 2 [ x ] \} , 0 \leq x \leq 2$, where $m$ is the number of points in $( 0,2 )$ where $f$ is not continuous and $n$ be the number of points in $( 0,2 )$, where $f$ is not differentiable. Then $( m + n ) ^ { 2 } + 2$ is equal to
(1) 2
(2) 11
(3) 6
(4) 3

If $\int_0^1 (x^{21} + x^{14} + x^7)(2x^{14} + 3x^7 + 6)^{1/7}\, dx = \frac{1}{l} \cdot 11^{m/n}$ where $l, m, n \in \mathbb{N}$, $m$ and $n$ are co-prime, then $l + m + n$ is equal to $\_\_\_\_$.

The integral $16 \int _ { 1 } ^ { 2 } \frac { d x } { x ^ { 3 } \left( x ^ { 2 } + 2 \right) ^ { 2 } }$ is equal to
(1) $\frac { 11 } { 6 } + \log _ { e } 4$
(2) $\frac { 11 } { 12 } + \log _ { e } 4$
(3) $\frac { 11 } { 12 } - \log _ { e } 4$
(4) $\frac { 11 } { 6 } - \log _ { e } 4$

If $\int _ { \frac { 1 } { 3 } } ^ { 3 } \left| \log _ { e } x \right| dx = \frac { m } { n } \log _ { e } \left( \frac { n ^ { 2 } } { e } \right)$, where m and n are coprime natural numbers, then $m ^ { 2 } + n ^ { 2 } - 5$ is equal to $\_\_\_\_$.

The value of $\int_{\pi/3}^{\pi/2} \frac{2 + 3\sin x}{\sin x(1 + \cos x)}\,dx$ is equal to
(1) $\frac{7}{2} - \sqrt{3} - \log_e\sqrt{3}$
(2) $-2 + 3\sqrt{3} + \log_e\sqrt{3}$
(3) $\frac{10}{3} - \sqrt{3} + \log_e\sqrt{3}$
(4) $\frac{10}{3} - \sqrt{3} - \log_e\sqrt{3}$

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) $2\log_e 3$

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

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