If $b_n = \int_0^{\frac{\pi}{2}} \frac{\cos^2 nx}{\sin x} dx$, $n \in \mathbb{N}$, then
(1) $b_3 - b_2, b_4 - b_3, b_5 - b_4$ are in an A.P. with common difference $-2$
(2) $\frac{1}{b_3 - b_2}, \frac{1}{b_4 - b_3}, \frac{1}{b_5 - b_4}$ are in an A.P. with common difference $2$
(3) $b_3 - b_2, b_4 - b_3, b_5 - b_4$ are in a G.P.
(4) $\frac{1}{b_3 - b_2}, \frac{1}{b_4 - b_3}, \frac{1}{b_5 - b_4}$ are in an A.P. with common difference $-2$

$I = \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( \frac { 8 \sin x - \sin 2 x } { x } \right) d x$. Then
(1) $\frac { \pi } { 2 } < I < \frac { 3 \pi } { 4 }$
(2) $\frac { \pi } { 5 } < I < \frac { 5 \pi } { 12 }$
(3) $\frac { 5 \pi } { 12 } < I < \frac { \sqrt { 2 } } { 3 } \pi$
(4) $\frac { 3 \pi } { 4 } < I < \pi$

The value of $k \in \mathrm {~N}$ for which the integral $I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 - x ^ { k } \right) ^ { n } d x , n \in \mathbb { N }$, satisfies $147 I _ { 20 } = 148 I _ { 21 }$ is
(1) 14
(2) 8
(3) 10
(4) 7

Let $n$ be a natural number,
$$\begin{aligned} & f _ { n } : [ n , n + 1 ) \rightarrow \left[ 0 , \frac { 1 } { 2 ^ { n } } \right) \\ & f _ { n } ( x ) = \frac { ( x - n ) ^ { 2 } } { 2 ^ { n } } \end{aligned}$$
The regions between the functions defined in this form and the x-axis are given shaded in the figure below.
Accordingly, what is the sum of the areas of all shaded regions in square units?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 8 } { 9 }$
E) $\frac { 11 } { 12 }$

For an increasing and continuous function f defined on the set of real numbers,
$$\begin{aligned} & f ( 0 ) = 2 \\ & f ( 1 ) = 3 \\ & f ( 2 ) = 4 \end{aligned}$$
equalities are given.
Accordingly, the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ could be which of the following?
A) 4 B) 4.5 C) 6 D) 7.5 E) 8

Let $m$ be a positive real number. In the rectangular coordinate plane, the region between the graph of a function $f$ defined on the closed interval $[-m, m]$ and the $x$-axis is divided into four regions and these regions are colored as shown in the figure. The areas of these regions, which are different from each other, are denoted by $A, B, C$ and $D$ as shown in the figure.
$$\int_{-m}^{m} |f(x)|\, dx = \int_{-m}^{m} f(x)\, dx + \int_{0}^{m} 2 \cdot f(x)\, dx$$
Given that, which of the following is the integral $\int_{-m}^{m} f(x)\, dx$ equal to?
A) $A + B$ B) $A + C$ C) $A + D$ D) $B + C$ E) $C + D$