We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
We set
$$P _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }, \quad ( 1 - y ) ^ { n } = \sum _ { k = 0 } ^ { n } b _ { k } y ^ { k }$$
with for all $k \in \llbracket 0 , n \rrbracket , a _ { k } \in \mathbb { Z }$ and $b _ { k } \in \mathbb { Z }$.
Let $n \in \mathbb { N } ^ { * }$. Justify the existence of
$$I _ { n } = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { ( 1 - y ) ^ { n } P _ { n } ( x ) } { 1 - x y } \mathrm {~d} x \mathrm {~d} y$$
and show that
$$I _ { n } = \sum _ { \substack { r , s = 0 \\ r \neq s } } ^ { n } a _ { r } b _ { s } J _ { r , s } + \sum _ { r = 0 } ^ { n } a _ { r } b _ { r } J _ { r , r }$$

Evaluate $\lim_{n \rightarrow \infty} \left\{\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{n+n}\right\}$.

Let $f(x) = \int_0^1 |t - x|\, t\, dt$ for $x \in [0,1]$. Find $f(x)$ and sketch its graph.

Show that $\int_1^n [u]([u]+1)f(u)\,du = 2\sum_{i=1}^{[n]} i \int_i^{i+1} f(u)\,du$ (or an equivalent integral identity involving the floor function).

Let $f$ be a differentiable function on $[0, 2\pi]$ with $f'(x)$ increasing. Show that $\int_0^{2\pi} f(x) \cos x \, dx \geq 0$.

Let $a$ and $\beta$ be two positive real numbers. For every integer $n > 0$, define $a_n = \int_{\beta}^{n} \frac{a}{u(u^{a}+2+u^{-a})} du$. Then $\lim_{n \to \infty} a_n$ is equal to
(a) $1/(1+\beta^{a})$
(b) $\beta^{a}/(1+\beta^{-a})$
(c) $\beta^{a}/(1+\beta^{a})$
(d) $\beta^{-a}/(1+\beta^{a})$

The equation $x^{2} + (b/a)x + (c/a) = 0$ has two real roots $\alpha$ and $\beta$. If $a > 0$, then the area under the curve $f(x) = x^{2} + (b/a)x + (c/a)$ between $\alpha$ and $\beta$ is
(a) $(b^{2} - 4ac)/2a$
(b) $(b^{2} - 4ac)^{3/2}/6a^{3}$
(c) $-(b^{2} - 4ac)^{3/2}/6a^{3}$
(d) $-(b^{2} - 4ac)/2a$

The value of $\lim_{n \to \infty} \sum_{r} \frac{6n}{9n^{2} - r^{2}}$ is
(a) 0
(b) $\log(3/2)$
(c) $\log(2/3)$
(d) $\log(2)$

Let $[ x ]$ denote the largest integer less than or equal to $x$. Then $\int_0^{n^{1/k}} \left[ x ^ { k } + n \right] dx$ equals
(a) $n ^ { 2 } + \sum_{i=1}^{n} i ^ { 1 / k }$
(b) $2 n ^ { ( 1 + k ) / k } - \sum_{i=1}^{n} i ^ { 1 / k }$
(c) $2 n ^ { ( 1 + k ) / k } - \sum_{i=1}^{n-1} i ^ { 1 / k }$
(d) None of these.

Let $f ( x ) = ( \tan x ) ^ { 3 / 2 } - 3 \tan x + \sqrt{\tan x}$. Consider the three integrals $I _ { 1 } = \int_0^1 f ( x ) \, dx$; $I _ { 2 } = \int_{0.3}^{1.3} f ( x ) \, dx$ and $I _ { 3 } = \int_{0.5}^{1.5} f ( x ) \, dx$. Then,
(a) $I _ { 1 } > I _ { 2 } > I _ { 3 }$
(b) $I _ { 2 } > I _ { 1 } > I _ { 3 }$
(c) $I _ { 3 } > I _ { 1 } > I _ { 2 }$
(d) $I _ { 1 } > I _ { 3 } > I _ { 2 }$

Let $f$ be a periodic function with period $1$, and let $g(t) = \int_0^t f(x)\,dx$. Define $h(t) = \lim_{n\to\infty} \dfrac{g(t+n)}{n}$. Which of the following is true about $h(t)$?
(A) $h(t)$ depends on $t$
(B) $h(t)$ is not defined for all $t$
(C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$
(D) None of the above

Let $n$ be a positive integer. Define $$f ( x ) = \min \{ | x - 1 | , | x - 2 | , \ldots , | x - n | \}.$$ Then $\int _ { 0 } ^ { n + 1 } f ( x ) d x$ equals
(A) $\frac { ( n + 4 ) } { 4 }$
(B) $\frac { ( n + 3 ) } { 4 }$
(C) $\frac { ( n + 2 ) } { 2 }$
(D) $\frac { ( n + 2 ) } { 4 }$

The value of the integral $$\int _ { \pi / 2 } ^ { 5 \pi / 2 } \frac { e ^ { \tan ^ { - 1 } ( \sin x ) } } { e ^ { \tan ^ { - 1 } ( \sin x ) } + e ^ { \tan ^ { - 1 } ( \cos x ) } } d x$$ equals (A) 1 (B) $\pi$ (C) $e$ (D) none of these

Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as given by the respective graphs.

If $a _ { n } = \left( 1 + \frac { 1 } { n ^ { 2 } } \right) \left( 1 + \frac { 2 ^ { 2 } } { n ^ { 2 } } \right) ^ { 2 } \left( 1 + \frac { 3 ^ { 2 } } { n ^ { 2 } } \right) ^ { 3 } \cdots \left( 1 + \frac { n ^ { 2 } } { n ^ { 2 } } \right) ^ { n }$, then $$\lim _ { n \rightarrow \infty } a _ { n } ^ { - 1 / n ^ { 2 } }$$ is
(A) 0
(B) 1
(C) $e$
(D) $\sqrt { e } / 2$

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac{3}{4}$
(B) $\frac{4}{3}$
(C) $\frac{3}{2}$
(D) $\frac{2}{3}$

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $(2, 3)$
(D) is greater than 3.

Let $n$ be a positive integer. Define $$f ( x ) = \min \{ | x - 1 | , | x - 2 | , \ldots , | x - n | \}$$ Then $\int _ { 0 } ^ { n + 1 } f ( x ) d x$ equals
(A) $\frac { ( n + 4 ) } { 4 }$
(B) $\frac { ( n + 3 ) } { 4 }$
(C) $\frac { ( n + 2 ) } { 2 }$
(D) $\frac { ( n + 2 ) } { 4 }$

Let $n$ be a positive integer. Define $$f ( x ) = \min \{ | x - 1 | , | x - 2 | , \ldots , | x - n | \}$$ Then $\int _ { 0 } ^ { n + 1 } f ( x ) d x$ equals
(A) $\frac { ( n + 4 ) } { 4 }$
(B) $\frac { ( n + 3 ) } { 4 }$
(C) $\frac { ( n + 2 ) } { 2 }$
(D) $\frac { ( n + 2 ) } { 4 }$

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac{3}{4}$
(B) $\frac{4}{3}$
(C) $\frac{3}{2}$
(D) $\frac{2}{3}$

Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is
(A) $\frac { 3 } { 4 }$
(B) $\frac { 4 } { 3 }$
(C) $\frac { 3 } { 2 }$
(D) $\frac { 2 } { 3 }$

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $(2, 3)$
(D) is greater than 3

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $( 2, 3 )$
(D) is greater than 3