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
$$I _ { n } = \sum _ { \substack { r , s = 0 \\ r \neq s } } ^ { n } a _ { r } b _ { s } J _ { r , s } + \sum _ { r = 0 } ^ { n } a _ { r } b _ { r } J _ { r , r }$$

Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Deduce the value of $\int _ { 0 } ^ { \infty } \frac { \sin t } { t } \mathrm {~d} t$.

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) $-(b^{2} - 4ac)/2a$

23. $\int \sqcap / 33 \pi / 4 \mathrm { dx } / ( 1 + \cos \mathrm { x } )$ is equal to :
(A) 2
(B) - 2
(C) $1 / 2$
(D) $- 1 / 2$

13. Consider an infinite geometric series with first term and common ratio $r$. If its sum is 4 and the second term is $3 / 4$, then :
(A) $a = 4 / 7 , r = 3 / 7$
(B) $\mathrm { a } = 2 , \mathrm { r } = 3 / 8$
(C) $\mathrm { a } = 3 / 2 , \mathrm { r } = 1 / 2$
(D) $\mathrm { a } = 3 , \mathrm { r } = 1 / 4$

If the function $\mathrm { f } : [ 0,4 ] - - > \mathrm { R }$ is differentiable then show that (i) For $\mathrm { a } , \mathrm { b } \hat { \mathrm { I } } ( 0,4 ) , ( \mathrm { f } ( 4 ) ) ^ { 2 } = \mathrm { f } ( \mathrm { a } ) \mathrm { f } ( \mathrm { b } )$ (ii) $\left. \int _ { 0 } ^ { 4 } \mathrm { f } ( \mathrm { t } ) \mathrm { dt } = 2 \left[ \alpha \mathrm { f } \left( \alpha ^ { 2 } \right) + \beta \mathrm { f } ( \beta ) ^ { 2 } \right] \right] \forall 0 < \alpha , \beta < 2$.

38. Match the following
(i) $\int _ { 0 } ^ { \pi / 2 } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \left( \cos \mathrm { x } \cot \mathrm { x } - \log ( \sin \mathrm { x } ) ^ { \sin \mathrm { x } } \right) \mathrm { dx }$
(A) 1
(ii) Area bounded by $- 4 y ^ { 2 } = x$ and $x - 1 = - 5 y ^ { 2 }$
(B) 0
(iii) Cosine of the angle of intersection of curves $y = 3 ^ { x - 1 } \log x$ and
$$y = x ^ { x } - 1 \text { is }$$
(C) $6 \ln 2$
(iv) Data could not be retrieved.
(D) $4 / 3$
Sol. (i) $I = \int _ { 0 } ^ { \pi / 2 } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \left( \cos \mathrm { x } \cdot \cot \mathrm { x } - \log ( \sin \mathrm { x } ) ^ { \sin \mathrm { x } } \right) \mathrm { dx }$
$$\Rightarrow \quad \mathrm { I } = \int _ { 0 } ^ { \pi / 2 } \frac { \mathrm {~d} } { \mathrm { dx } } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \mathrm { dx } = 1 .$$
(ii) The points of intersection of $- 4 y ^ { 2 } = x$ and $x - 1 = - 5 y ^ { 2 }$ is $( - 4 , - 1 )$ and $( - 4,1 )$
Hence required area $= 2 \left[ \mid \int _ { 0 } ^ { 1 } \left( 1 - 5 y ^ { 2 } \right) d y - \int _ { 0 } ^ { 1 } - 4 y ^ { 2 } d y \right] \left\lvert \, = \frac { 4 } { 3 } \right.$.
(iii) The point of intersection of $y = 3 ^ { x - 1 } \log x$ and $y = x ^ { x } - 1$ is $( 1,0 )$
Hence $\frac { d y } { d x } = \frac { 3 ^ { x - 1 } } { x } + 3 ^ { x - 1 } \log 3 \cdot \log x . \left. \quad \frac { d y } { d x } \right| _ { ( 1,0 ) } = 1$ for $\mathrm { y } = \mathrm { x } ^ { \mathrm { x } } - \left. 1 \cdot \frac { \mathrm { dy } } { \mathrm { dx } } \right| _ { ( 1,0 ) } = 1$ If $\theta$ is the angle between the curve then $\tan \theta = 0 \Rightarrow \cos \theta = 1$.
(iv) $\frac { \mathrm { dy } } { \mathrm { dx } } = \left( \frac { 2 } { \mathrm { x } + \mathrm { y } } \right)$ $\Rightarrow \frac { \mathrm { dx } } { \mathrm { dy } } - \frac { \mathrm { x } } { 2 } = \frac { \mathrm { y } } { 2 }$ $\Rightarrow \quad \mathrm { xe } ^ { - \mathrm { y } / 2 } = \frac { 1 } { 2 } \int \mathrm { y } \cdot \mathrm { e } ^ { - \mathrm { y } / 2 } \mathrm { dy }$ $\Rightarrow \quad x + y + 2 = k ^ { \mathrm { y } / 2 } = 3 \mathrm { e } ^ { \mathrm { y } / 2 }$.

The value(s) of $\int _ { 0 } ^ { 1 } \frac { x ^ { 4 } ( 1 - x ) ^ { 4 } } { 1 + x ^ { 2 } } d x$ is (are)
A) $\frac { 22 } { 7 } - \pi$
B) $\frac { 2 } { 105 }$
C) 0
D) $\frac { 71 } { 15 } - \frac { 3 \pi } { 2 }$

Let $f ( x ) = 7 \tan ^ { 8 } x + 7 \tan ^ { 6 } x - 3 \tan ^ { 4 } x - 3 \tan ^ { 2 } x$ for all $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the correct expression(s) is(are)
(A) $\quad \int _ { 0 } ^ { \pi / 4 } x f ( x ) d x = \frac { 1 } { 12 }$
(B) $\quad \int _ { 0 } ^ { \pi / 4 } f ( x ) d x = 0$
(C) $\int _ { 0 } ^ { \pi / 4 } x f ( x ) d x = \frac { 1 } { 6 }$
(D) $\int _ { 0 } ^ { \pi / 4 } f ( x ) d x = 1$

If $\int _ { 1 } ^ { 3 } x ^ { 2 } F ^ { \prime \prime } ( x ) d x = - 12$ and $\int _ { 1 } ^ { 3 } x ^ { 3 } F ^ { \prime \prime } ( x ) d x = 40$, then the correct expression(s) is(are)
(A) $9 f ^ { \prime } ( 3 ) + f ^ { \prime } ( 1 ) - 32 = 0$
(B) $\int _ { 1 } ^ { 3 } f ( x ) d x = 12$
(C) $9 f ^ { \prime } ( 3 ) - f ^ { \prime } ( 1 ) + 32 = 0$
(D) $\int _ { 1 } ^ { 3 } f ( x ) d x = - 12$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$, $f\left(\frac{\pi}{2}\right) = 3$ and $f'(0) = 1$. If $$g(x) = \int_x^{\frac{\pi}{2}} \left[f'(t)\operatorname{cosec} t - \cot t\operatorname{cosec} t\, f(t)\right] dt$$ for $x \in \left(0, \frac{\pi}{2}\right]$, then $\lim_{x \rightarrow 0} g(x) =$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi) = -6$.
If $F: [0, \pi] \rightarrow \mathbb{R}$ is defined by $F(x) = \int_{0}^{x} f(t)\, dt$, and if $$\int_{0}^{\pi} \left(f'(x) + F(x)\right) \cos x\, dx = 2$$ then the value of $f(0)$ is $\_\_\_\_$

Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{16 S_1}{\pi}$ is ____.

Let $g_i : [\pi/8, 3\pi/8] \to \mathbb{R}$, $i = 1, 2$, and $f : [\pi/8, 3\pi/8] \to \mathbb{R}$ be functions such that $$g_1(x) = 1, \quad g_2(x) = |4x - \pi|, \quad f(x) = \sin^2 x,$$ for all $x \in [\pi/8, 3\pi/8]$. Define $$S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx, \quad i = 1, 2.$$
The value of $\frac{48 S_2}{\pi^2}$ is ____.

Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 16 S _ { 1 } } { \pi }$ is $\_\_\_\_$.

Let $g _ { i } : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R } , i = 1,2$, and $f : \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right] \rightarrow \mathbb { R }$ be functions such that $$g _ { 1 } ( x ) = 1 , g _ { 2 } ( x ) = | 4 x - \pi | \text { and } f ( x ) = \sin ^ { 2 } x , \text { for all } x \in \left[ \frac { \pi } { 8 } , \frac { 3 \pi } { 8 } \right]$$ Define $$S _ { i } = \int _ { \frac { \pi } { 8 } } ^ { \frac { 3 \pi } { 8 } } f ( x ) \cdot g _ { i } ( x ) d x , \quad i = 1,2$$ The value of $\frac { 48 S _ { 2 } } { \pi ^ { 2 } }$ is $\_\_\_\_$.

The greatest integer less than or equal to
$$\int _ { 1 } ^ { 2 } \log _ { 2 } \left( x ^ { 3 } + 1 \right) d x + \int _ { 1 } ^ { \log _ { 2 } 9 } \left( 2 ^ { x } - 1 \right) ^ { \frac { 1 } { 3 } } d x$$
is $\_\_\_\_$ .

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $2 \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x - \int _ { 0 } ^ { \frac { \pi } { 2 } } g ( x ) d x$ is $\_\_\_\_$ .

Let $f : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0,1 ]$ be the function defined by $f ( x ) = \sin ^ { 2 } x$ and let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow [ 0 , \infty )$ be the function defined by $g ( x ) = \sqrt { \frac { \pi x } { 2 } - x ^ { 2 } }$. The value of $\frac { 16 } { \pi ^ { 3 } } \int _ { 0 } ^ { \frac { \pi } { 2 } } f ( x ) g ( x ) d x$ is $\_\_\_\_$ .

The integral $\int _ { 0 } ^ { \pi } \sqrt { 1 + 4 \sin ^ { 2 } \frac { x } { 2 } - 4 \sin \frac { x } { 2 } } \, d x$ equals
(1) $4 \sqrt { 3 } - 4$
(2) $4 \sqrt { 3 } - 4 - \frac { \pi } { 3 }$
(3) $\pi - 4$
(4) $\frac { 2 \pi } { 3 } - 4 - 4 \sqrt { 3 }$

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}$ is equal to:
(1) $-1$
(2) $-2$
(3) $2$
(4) $4$

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1+\cos x}$ is equal to: (1) $-1$ (2) $-2$ (3) $2$ (4) $4$

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

The integral $\displaystyle\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$ is equal to
(1) $-2$
(2) $2$
(3) $4$
(4) $-1$

The integral $\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 4 } } \frac { 8 \cos 2 x } { ( \tan x + \cot x ) ^ { 3 } } d x$ equals
(1) $\frac { 13 } { 256 }$
(2) $\frac { 15 } { 64 }$
(3) $\frac { 13 } { 32 }$
(4) $\frac { 15 } { 128 }$