The function $f$ is defined by $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ for $- 5 \leq x \leq 5$.
(a) Find $f ^ { \prime } ( x )$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = - 3$.
(c) Let $g$ be the function defined by $g ( x ) = \begin{cases} f ( x ) & \text { for } - 5 \leq x \leq - 3 \\ x + 7 & \text { for } - 3 < x \leq 5 . \end{cases}$
Is $g$ continuous at $x = - 3$ ? Use the definition of continuity to explain your answer.
(d) Find the value of $\int _ { 0 } ^ { 5 } x \sqrt { 25 - x ^ { 2 } } d x$.

Consider the family of functions $f ( x ) = \frac { 1 } { x ^ { 2 } - 2 x + k }$, where $k$ is a constant.
(a) Find the value of $k$, for $k > 0$, such that the slope of the line tangent to the graph of $f$ at $x = 0$ equals 6.
(b) For $k = - 8$, find the value of $\int _ { 0 } ^ { 1 } f ( x ) \, dx$.
(c) For $k = 1$, find the value of $\int _ { 0 } ^ { 2 } f ( x ) \, dx$ or show that it diverges.

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 $m_{0}$.

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) $- b e ^ { a }$

5. Evaluate $\int \sin - 1 ( ( 2 \mathrm { x } + 2 ) / \sqrt { } ( 4 \mathrm { x } 2 + 8 \mathrm { x } + 13 ) ) d \mathrm { x }$.

9. For any natural number m , evaluate
$$\int \left( x ^ { 3 m } + x ^ { 2 m } + x ^ { m } \right) \left( 2 x ^ { 2 m } + 3 x ^ { m } + 6 \right) ^ { \frac { 1 } { m } } d x , x > 0$$

1. Let

$$\begin{aligned} & \text { Let } f ( x ) = \left\{ \begin{array} { c l } x + a , & x < 0 \\ | x - 1 | , & x \geq 0 , \end{array} \right. \\ & \text { And } g ( x ) = \left\{ \begin{array} { c l } x + 1 & \text { if } x < 0 \\ ( x - 1 ) ^ { 2 } + b & \text { if } x \geq 0 \end{array} \right. \end{aligned}$$
where a and b aneegetive real numbers. Determine the composite function gof. (If (gof) (x) is continuous for all real x , determine the values of a and b . Further, for these values of a and b , is gof differentiable at $\mathrm { x } = 0$ ? Justify your answer.

2. $\quad \int \frac { x ^ { 2 } - 1 } { x ^ { 3 } \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } d x$ is equal to
(A) $\frac { \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } { x ^ { 2 } } + c$
(B) $\frac { \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } { x ^ { 3 } } + c$
(C) $\frac { \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } { x } + c$
(D) $\frac { \sqrt { 2 x ^ { 4 } - 2 x ^ { 2 } + 1 } } { 2 x ^ { 2 } } + c$
Sol. (D)
$$\int \frac { \left( \frac { 1 } { x ^ { 3 } } - \frac { 1 } { x ^ { 5 } } \right) d x } { \sqrt { 2 - \frac { 2 } { x ^ { 2 } } + \frac { 1 } { x ^ { 4 } } } }$$
Let $2 - \frac { 2 } { x ^ { 2 } } + \frac { 1 } { x ^ { 4 } } = z \Rightarrow \frac { 1 } { 4 } \int \frac { d z } { \sqrt { z } }$ $\Rightarrow \quad \frac { 1 } { 2 } \times \sqrt { \mathrm { z } } + \mathrm { c } \Rightarrow \frac { 1 } { 2 } \sqrt { 2 - \frac { 2 } { \mathrm { x } ^ { 2 } } + \frac { 1 } { \mathrm { x } ^ { 4 } } } + \mathrm { c }$.

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 the integral $\int _ { 1 } ^ { 2 } \left( \frac { t ^ { 4 } + 1 } { t ^ { 6 } + 1 } \right) d t$ is : (1) $\tan ^ { - 1 } \frac { 1 } { 2 } + \frac { 1 } { 3 } \tan ^ { - 1 } 8 - \frac { \pi } { 3 }$ (2) $\tan ^ { - 1 } 2 - \frac { 1 } { 3 } \tan ^ { - 1 } 8 + \frac { \pi } { 3 }$ (3) $\tan ^ { - 1 } 2 + \frac { 1 } { 3 } \tan ^ { - 1 } 8 - \frac { \pi } { 3 }$ (4) $\tan ^ { - 1 } \frac { 1 } { 2 } - \frac { 1 } { 3 } \tan ^ { - 1 } 8 + \frac { \pi } { 3 }$

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) $\frac{10}{3} - \sqrt{3} - \log_e\sqrt{3}$