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

Consider the following integral $I _ { n } ( \alpha )$ for $\alpha \geq 1$ and $n > 0$.
$$I _ { n } ( \alpha ) = \int _ { \frac { 1 } { n } } ^ { n } \frac { f ( \alpha x ) - f ( x ) } { x } \mathrm {~d} x$$
Assume that a real-valued function $f ( x )$ is continuous and differentiable on $x \geq 0$, its derivative is continuous, and $\lim _ { x \rightarrow \infty } f ( x ) = 0$. Answer the following questions.
(1) Define $J _ { n } ( \alpha ) = \frac { \mathrm { d } I _ { n } ( \alpha ) } { \mathrm { d } \alpha }$. Show that $J _ { n } ( \alpha ) = \frac { 1 } { \alpha } \left( f ( \alpha n ) - f \left( \frac { \alpha } { n } \right) \right)$.
You can use the fact that the integration and the differentiation commute in this context.
(2) Define $I ( \alpha ) = \lim _ { n \rightarrow \infty } I _ { n } ( \alpha )$. Show that $\lim _ { n \rightarrow \infty } J _ { n } ( \beta )$ exists for any $\beta \in [ 1 , \alpha ]$ and it uniformly converges on $[ 1 , \alpha ]$, and show that
$$I ( \alpha ) = \int _ { 1 } ^ { \alpha } \left( \lim _ { n \rightarrow \infty } J _ { n } ( \beta ) \right) \mathrm { d } \beta$$
(3) Obtain $I ( \alpha )$.
(4) Calculate the following integral. Note that $p > q > 0$.
$$\int _ { 0 } ^ { \infty } \frac { e ^ { - p x } \cos ( p x ) - e ^ { - q x } \cos ( q x ) } { x } \mathrm {~d} x$$

1
(1) For a positive integer $k$, let $A_k = \displaystyle\int_{\sqrt{k\pi}}^{\sqrt{(k+1)\pi}} |\sin(x^2)|\,dx$. Prove that the following inequality holds: $$\frac{1}{\sqrt{(k+1)\pi}} \leqq A_k \leqq \frac{1}{\sqrt{k\pi}}$$
(2) For a positive integer $n$, let $B_n = \dfrac{1}{\sqrt{n}}\displaystyle\int_{\sqrt{n\pi}}^{\sqrt{2n\pi}} |\sin(x^2)|\,dx$. Find the limit $\displaystyle\lim_{n\to\infty} B_n$.
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$$\int_{0}^{\frac{\pi}{3}} \frac{\sin x}{\cos^{2} x}\, dx$$
What is the value of the integral?
A) 2
B) 1
C) 0
D) $-1$
E) $-2$

$$\int_{0}^{4} \frac{6x}{\sqrt{2x+1}}\, dx$$
What is the value of the integral?
A) 12
B) 15
C) 18
D) 20
E) 24

In the integral $\int \frac { \ln \sqrt { x } } { \sqrt { x } } d x$, if the substitution $u = \sqrt { x }$ is made, which of the following integrals is obtained?
A) $\int \ln u \, d u$
B) $\int 2 \ln u \, d u$
C) $\int \frac { \ln u } { u } d u$
D) $\int \frac { \ln u } { 2 u } d u$
E) $\int u \ln u \, d u$

$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 2 x \cdot \cot x \, d x$$
What is the value of this integral?
A) $\frac { \pi + 1 } { 2 }$
B) $\frac { \pi + 1 } { 3 }$
C) $\frac { \pi + 2 } { 4 }$
D) $\frac { \pi - 1 } { 6 }$
E) $\frac { \pi - 2 } { 6 }$

$$\int _ { 4 } ^ { 9 } \frac { \sqrt { x } } { x - 1 } d x$$
If the substitution $\mathbf { u } = \sqrt { \mathbf { x } }$ is made in the integral, which of the following integrals is obtained?
A) $\int _ { 4 } ^ { 9 } \frac { u } { u ^ { 2 } - 1 } d u$
B) (missing option)
C) $\int _ { 2 } ^ { 3 } \frac { u } { 2 \left( u ^ { 2 } - 1 \right) } d u$
D) $\int _ { 2 } ^ { 3 } \frac { 2 u ^ { 2 } } { u ^ { 2 } - 1 } d u$
E) $\int _ { 2 } ^ { 3 } \frac { u } { u ^ { 2 } - 1 } d u$

$$\int _ { 2 } ^ { 3 } \frac { 2 x ^ { 2 } } { x ^ { 2 } - 1 } d x$$
What is the value of the integral?
A) $1 + \ln \left( \frac { 4 } { 3 } \right)$
B) $1 + \ln \left( \frac { 9 } { 2 } \right)$
C) $2 + \ln \left( \frac { 3 } { 2 } \right)$
D) $2 + \ln \left( \frac { 5 } { 3 } \right)$
E) $3 + \ln \left( \frac { 1 } { 3 } \right)$

$$\int _ { 4 } ^ { 9 } \frac { 3 x - 3 } { \sqrt { x } + 1 } d x$$
What is the value of the integral?
A) 13
B) 18
C) 23
D) 28
E) 33

$\int \sqrt { 1 + e^{x^{2}} } \, d x$\ In the integral, if the substitution $u = \sqrt { 1 + e ^ { x } }$ is made, which of the following integrals is obtained?\ A) $\int \frac { 2 u } { u ^ { 2 } + 1 } d u$\ B) $\int \frac { u ^ { 2 } } { u ^ { 2 } + 1 } d u$\ C) $\int \frac { 1 } { u ^ { 2 } - 1 } d u$\ D) $\int \frac { u } { u ^ { 2 } - 1 } d u$\ E) $\int \frac { 2 u ^ { 2 } } { u ^ { 2 } - 1 } d u$

$$\int \frac { ( 3 \sqrt { x } + 2 ) ^ { 5 } } { \sqrt { x } } d x$$
Which of the following is this integral equal to? (c is an arbitrary constant.)
A) $\frac { 1 } { 18 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
B) $\frac { 1 } { 9 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
C) $\frac { 2 } { 9 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
D) $\frac { 1 } { 3 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
E) $\frac { 2 } { 3 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$

$$\int_{1}^{2} (x+2) \cdot \sqrt[3]{x^{2} + 4x - 4}\, dx$$
What is the value of this integral?
A) $\dfrac{45}{8}$ B) $\dfrac{47}{8}$ C) $\dfrac{49}{8}$ D) $\dfrac{45}{4}$ E) $\dfrac{47}{4}$