Let
$$I = \int \frac { e ^ { x } } { e ^ { 4 x } + e ^ { 2 x } + 1 } d x , \quad J = \int \frac { e ^ { - x } } { e ^ { - 4 x } + e ^ { - 2 x } + 1 } d x$$
Then, for an arbitrary constant $C$, the value of $J - I$ equals
(A) $\frac { 1 } { 2 } \log \left( \frac { e ^ { 4 x } - e ^ { 2 x } + 1 } { e ^ { 4 x } + e ^ { 2 x } + 1 } \right) + C$
(B) $\frac { 1 } { 2 } \log \left( \frac { e ^ { 2 x } + e ^ { x } + 1 } { e ^ { 2 x } - e ^ { x } + 1 } \right) + C$
(C) $\frac { 1 } { 2 } \log \left( \frac { e ^ { 2 x } - e ^ { x } + 1 } { e ^ { 2 x } + e ^ { x } + 1 } \right) + C$
(D) $\frac { 1 } { 2 } \log \left( \frac { e ^ { 4 x } + e ^ { 2 x } + 1 } { e ^ { 4 x } - e ^ { 2 x } + 1 } \right) + C$

For $x \in \mathbb { R }$, let $\tan ^ { - 1 } ( x ) \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the minimum value of the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { x \tan ^ { - 1 } x } \frac { e ^ { ( t - \cos t ) } } { 1 + t ^ { 2023 } } d t$ is

Let $\alpha$ and $\beta$ be the real numbers such that
$$\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 3 } } \left( \frac { \alpha } { 2 } \int _ { 0 } ^ { x } \frac { 1 } { 1 - t ^ { 2 } } d t + \beta x \cos x \right) = 2$$
Then the value of $\alpha + \beta$ is $\_\_\_\_$ .

$\int \frac { \sin ^ { 8 } x - \cos ^ { 8 } x } { \left( 1 - 2 \sin ^ { 2 } x \cos ^ { 2 } x \right) } d x$ is equal to
(1) $- \frac { 1 } { 2 } \sin 2 x + c$
(2) $- \sin ^ { 2 } x + c$
(3) $- \frac { 1 } { 2 } \sin x + c$
(4) $\frac { 1 } { 2 } \sin 2 x + c$

The integral $\int \frac { 2 x ^ { 3 } - 1 } { x ^ { 4 } + x } d x$ is equal to (here $C$ is a constant of integration)
(1) $\frac { 1 } { 2 } \ln \frac { | x ^ { 3 } + 1 | } { x ^ { 2 } } + C$
(2) $\frac { 1 } { 2 } \ln \frac { ( x ^ { 3 } + 1 ) ^ { 2 } } { | x ^ { 3 } | } + C$
(3) $\ln \frac { | x ^ { 3 } + 1 | } { x ^ { 2 } } + C$
(4) $\ln \frac { | x ^ { 3 } + 1 | } { x ^ { 3 } } + C$

Let $I_n = \int \tan^n x\, dx$ $(n > 1)$. If $I_4 + I_6 = a\tan^5 x + bx^5 + c$, then the ordered pair $(a, b)$ is equal to
(1) $\left(-\dfrac{1}{5}, 1\right)$
(2) $\left(\dfrac{1}{5}, 0\right)$
(3) $\left(\dfrac{1}{5}, -1\right)$
(4) $\left(-\dfrac{1}{5}, 0\right)$

The integral $\int \sqrt { 1 + 2 \cot x ( \operatorname { cosec } x + \cot x ) } d x , \left( 0 < x < \frac { \pi } { 2 } \right)$ is equal to
(1) $2 \log \left| \sin \frac { x } { 2 } \right| + c$
(2) $4 \log \left| \sin \frac { x } { 2 } \right| + c$
(3) $4 \log \left| \cos \frac { x } { 2 } \right| + c$
(4) $2 \log \left| \cos \frac { x } { 2 } \right| + c$

Let $f : (0,2) \rightarrow R$ be a twice differentiable function such that $f''(x) > 0$, for all $x \in (0,2)$. If $\phi(x) = f(x) + f(2-x)$, then $\phi$ is
(1) decreasing on $(0,2)$
(2) increasing on $(0,2)$
(3) increasing on $(0,1)$ and decreasing on $(1,2)$
(4) decreasing on $(0,1)$ and increasing on $(1,2)$

If $\int e ^ { \sec x } \left( \sec x \tan x f ( x ) + \left( \sec x \tan x + \sec ^ { 2 } x \right) \right) d x = e ^ { \sec x } f ( x ) + C$, then a possible choice of $f ( x )$ is:
(1) $\sec x - \tan x - \frac { 1 } { 2 }$
(2) $\sec x + \tan x + \frac { 1 } { 2 }$
(3) $x \sec x + \tan x + \frac { 1 } { 2 }$
(4) $\sec x + x \tan x - \frac { 1 } { 2 }$

If $\int \frac { d \theta } { \cos ^ { 2 } \theta ( \tan 2 \theta + \sec 2 \theta ) } = \lambda \tan \theta + 2 \log _ { e } | f ( \theta ) | + C$ where $C$ is a constant of integration, then the ordered pair $( \lambda , f ( \theta ) )$ is equal to:
(1) $( 1,1 - \tan \theta )$
(2) $( - 1,1 - \tan \theta )$
(3) $( - 1,1 + \tan \theta )$
(4) $( 1,1 + \tan \theta )$

If $\int \frac{\cos\theta}{5 + 7\sin\theta - 2\cos^2\theta}\,d\theta = A\log_e|B(\theta)| + C$, where $C$ is a constant of integration, then $\frac{B(\theta)}{A}$ can be:
(1) $\frac{2\sin\theta+1}{\sin\theta+3}$
(2) $\frac{2\sin\theta+1}{5(\sin\theta+3)}$
(3) $\frac{5(\sin\theta+3)}{2\sin\theta+1}$
(4) $\frac{5(2\sin\theta+1)}{\sin\theta+3}$

Let $g : ( 0 , \infty ) \rightarrow R$ be a differentiable function such that $\int \left( \frac { x \cos x - \sin x } { e ^ { x } + 1 } + \frac { g(x) ( e ^ { x } + 1 ) - x e ^ { x } } { ( e ^ { x } + 1 ) ^ { 2 } } \right) \mathrm { d } x = \frac { x g(x) } { e ^ { x } + 1 } + C$, for all $x > 0$, where $C$ is an arbitrary constant. Then
(1) $g$ is decreasing in $\left( 0 , \frac { \pi } { 4 } \right)$
(2) $g - g ^ { \prime }$ is increasing in $\left( 0 , \frac { \pi } { 2 } \right)$
(3) $g ^ { \prime }$ is increasing in $\left( 0 , \frac { \pi } { 4 } \right)$
(4) $g + g ^ { \prime }$ is increasing in $\left( 0 , \frac { \pi } { 2 } \right)$

Let $I ( x ) = \int \frac { 6 } { \sin ^ { 2 } x ( 1 - \cot x ) ^ { 2 } } d x$. If $I ( 0 ) = 3$, then $I \left( \frac { \pi } { 12 } \right)$ is equal to
(1) $2 \sqrt { 3 }$
(2) $\sqrt { 3 }$
(3) $3 \sqrt { 3 }$
(4) $6 \sqrt { 3 }$

If $\int \frac { 1 } { \mathrm { a } ^ { 2 } \sin ^ { 2 } x + \mathrm { b } ^ { 2 } \cos ^ { 2 } x } \mathrm {~d} x = \frac { 1 } { 12 } \tan ^ { - 1 } ( 3 \tan x ) +$ constant, then the maximum value of $\mathrm { a } \sin x + \mathrm { b } \cos x$, is:
(1) $\sqrt { 40 }$
(2) $\sqrt { 41 }$
(3) $\sqrt { 39 }$
(4) $\sqrt { 42 }$

If $f ( t ) = \int _ { 0 } ^ { \pi } \frac { 2 x \mathrm {~d} x } { 1 - \cos ^ { 2 } \mathrm { t } \sin ^ { 2 } x } , 0 < \mathrm { t } < \pi$, then the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \pi ^ { 2 } \mathrm { dt } } { f ( \mathrm { t } ) }$ equals $\_\_\_\_$

If $\int \mathrm { e } ^ { x } \left( \frac { x \sin ^ { - 1 } x } { \sqrt { 1 - x ^ { 2 } } } + \frac { \sin ^ { - 1 } x } { \left( 1 - x ^ { 2 } \right) ^ { 3 / 2 } } + \frac { x } { 1 - x ^ { 2 } } \right) \mathrm { d } x = \mathrm { g } ( x ) + \mathrm { C }$, where C is the constant of integration, then $g \left( \frac { 1 } { 2 } \right)$ equals :
(1) $\frac { \pi } { 4 } \sqrt { \frac { e } { 3 } }$
(2) $\frac { \pi } { 6 } \sqrt { \frac { e } { 3 } }$
(3) $\frac { \pi } { 4 } \sqrt { \frac { e } { 2 } }$
(4) $\frac { \pi } { 6 } \sqrt { \frac { e } { 2 } }$

$$\int_{0}^{\frac{\pi}{3}} \frac{\sin x}{\cos^{2} x}\, dx$$
What is the value of the integral?
A) 2
B) 1
C) 0
D) $-1$
E) $-2$

$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 2 x \cdot \cot x \, d x$$
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) $\frac { \pi - 2 } { 6 }$