Let $F ( x ) = f ( x ) + f \left( \frac { 1 } { x } \right)$, where $f ( x ) = \int _ { 1 } ^ { x } \frac { \log t } { 1 + t } d t$. Then $F ( e )$ equals
(1) $\frac { 1 } { 2 }$
(2) 0
(3) 1
(4) 2

Statement-I: The value of the integral $\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to $\frac{\pi}{6}$. Statement-II: $\int_a^b f(x)\, dx = \int_a^b f(a + b - x)\, dx$.
(1) Statement-I is true; Statement-II is false.
(2) Statement-I is false; Statement-II is true.
(3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.

The integral $\int _ { 2 } ^ { 4 } \frac { \log x ^ { 2 } } { \log x ^ { 2 } + \log ( 6 - x ) ^ { 2 } } d x$ is equal to
(1) 6
(2) 2
(3) 4
(4) 1

The value of the integral $\int _ { 4 } ^ { 10 } \frac { \left[ x ^ { 2 } \right] } { \left[ x ^ { 2 } - 28 x + 196 \right] + \left[ x ^ { 2 } \right] } d x$, where $[ x ]$ denotes the greatest integer less than or equal to $x$, is
(1) $\frac { 1 } { 3 }$
(2) 6
(3) 7
(4) 3

The values of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { 2 } x } { 1 + 2 ^ { x } } d x$ is
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 8 }$
(3) $\frac { \pi } { 2 }$
(4) $4 \pi$

The value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin ^ { 4 } x \left( 1 + \ln \left( \frac { 2 + \sin x } { 2 - \sin x } \right) \right) d x$ is
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 8 } \pi$
(3) 0
(4) $\frac { 3 } { 16 } \pi$

The value of the integral
$$\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin ^ { 4 } x \left( 1 + \log \left( \frac { 2 + \sin x } { 2 - \sin x } \right) \right) d x$$
is
(1) $\frac { 3 } { 16 } \pi$
(2) 0
(3) $\frac { 3 } { 8 } \pi$
(4) $\frac { 3 } { 4 }$

If $f(x) = \frac{2 - x\cos x}{2 + x\cos x}$ and $g(x) = \log_e x$, then the value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} g(f(x))\, dx$ is
(1) $\log_e e$
(2) $\log_e 2$
(3) $\log_e 1$
(4) $\log_e 3$

The value of $\int _ { 0 } ^ { \pi / 2 } \frac { \sin ^ { 3 } x } { \sin x + \cos x } d x$ is:
(1) $\frac { \pi - 1 } { 2 }$
(2) $\frac { \pi - 2 } { 8 }$
(3) $\frac { \pi - 1 } { 4 }$
(4) $\frac { \pi - 2 } { 4 }$

If $f(a + b + 1 - x) = f(x)$, for all $x$, where $a$ and $b$ are fixed positive real numbers, then $\frac { 1 } { a + b } \int _ { a } ^ { b } x (f(x) + f(x + 1)) d x$ is equal to
(1) $\int _ { a - 1 } ^ { b - 1 } f(x + 1) d x$
(2) $\int _ { a - 1 } ^ { b - 1 } f(x) d x$
(3) $\int _ { a + 1 } ^ { b + 1 } f(x) d x$
(4) $\int _ { a + 1 } ^ { b + 1 } f(x + 1) d x$

The value of $\int _ { 0 } ^ { 2 \pi } \frac { x \sin ^ { 8 } x } { \sin ^ { 8 } x + \cos ^ { 8 } x } d x$ is equal to:
(1) $2 \pi$
(2) $2 \pi ^ { 2 }$
(3) $\pi ^ { 2 }$
(4) $4 \pi$

The value of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { 1 } { 1 + e ^ { \sin x } } d x$ is :
(1) $\frac { \pi } { 4 }$
(2) $\pi$
(3) $\frac { \pi } { 2 }$
(4) $\frac { 3 \pi } { 2 }$

For $x > 0$, if $f ( x ) = \int _ { 1 } ^ { x } \frac { \log _ { e } t } { ( 1 + t ) } d t$, then $f ( e ) + f \left( \frac { 1 } { e } \right)$ is equal to
(1) 0
(2) $\frac { 1 } { 2 }$
(3) - 1
(4) 1

Which of the following statement is correct for the function $g ( \alpha )$ for $\alpha \in R$ such that $g ( \alpha ) = \int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \frac { \sin ^ { \alpha } x } { \cos ^ { \alpha } x + \sin ^ { \alpha } x } d x$
(1) $g ( \alpha )$ is a strictly increasing function
(2) $g ( \alpha )$ has an inflection point at $\alpha = - \frac { 1 } { 2 }$
(3) $g ( \alpha )$ is a strictly decreasing function
(4) $g ( \alpha )$ is an even function

The value of $\int _ { - \pi / 2 } ^ { \pi / 2 } \left( \frac { 1 + \sin ^ { 2 } x } { 1 + \pi ^ { \sin x } } \right) d x$ is

Let $f : R \rightarrow R$ be a continuous function such that $f ( x ) + f ( x + 1 ) = 2$ for all $x \in R$. If $I _ { 1 } = \int _ { 0 } ^ { 8 } f ( x ) d x$ and $I _ { 2 } = \int _ { - 1 } ^ { 3 } f ( x ) d x$, then the value of $I _ { 1 } + 2 I _ { 2 }$ is equal to $\_\_\_\_$.

Let $g ( t ) = \int _ { - \pi / 2 } ^ { \pi / 2 } \left( \cos \frac { \pi } { 4 } t + f ( x ) \right) d x$, where $f ( x ) = \log _ { e } \left( x + \sqrt { x ^ { 2 } + 1 } \right) , x \in R$. Then which one of the following is correct?
(1) $g ( 1 ) = g ( 0 )$
(2) $\sqrt { 2 } g ( 1 ) = g ( 0 )$
(3) $g ( 1 ) = \sqrt { 2 } g ( 0 )$
(4) $g ( 1 ) + g ( 0 ) = 0$

The value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { d x } { \left( 1 + e ^ { x } \right) \left( \sin ^ { 6 } x + \cos ^ { 6 } x \right) }$ is equal to
(1) $2 \pi$
(2) 0
(3) $\pi$
(4) $\frac { \pi } { 2 }$

The value of $\int _ { 0 } ^ { \pi } \frac { e ^ { \cos x } \sin x } { ( 1 + \cos ^ { 2 } x )( e ^ { \cos x } + e ^ { - \cos x } ) } \mathrm { d } x$ is equal to

If $f ( \alpha ) = \int _ { 1 } ^ { \alpha } \frac { \log _ { 10 } t } { 1 + t } d t , \alpha > 0$, then $f \left( e ^ { 3 } \right) + f \left( e ^ { - 3 } \right)$ is equal to
(1) 9
(2) $\frac { 9 } { 2 }$

The value of $\int_0^1 (2x^3 - 3x^2 - x + 1)^{\frac{1}{3}} dx$ is equal to:
(1) 0
(2) 1
(3) 2
(4) $-1$

If the value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \left( \frac { x ^ { 2 } \cos x } { 1 + \pi ^ { x } } + \frac { 1 + \sin ^ { 2 } x } { 1 + e ^ { ( \sin x ) ^ { 2023 } } } \right) d x = \frac { \pi } { 4 } ( \pi + a ) - 2$, then the value of $a$ is
(1) 3
(2) $- \frac { 3 } { 2 }$
(3) 2
(4) $\frac { 3 } { 2 }$

Let $\mathrm { fx } = \int _ { 0 } ^ { \mathrm { x } } \mathrm { gt } \log _ { \mathrm { e } } \frac { 1 - \mathrm { t } } { 1 + \mathrm { t } } \mathrm { dt }$, where g is a continuous odd function. If $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \mathrm { fx } + \frac { \mathrm { x } ^ { 2 } \cos \mathrm { x } } { 1 + \mathrm { e } ^ { \mathrm { x } } } \mathrm { dx } = \frac { \pi ^ { 2 } } { \alpha } - \alpha$, then $\alpha$ is equal to $\_\_\_\_$.

Let for $f ( x ) = 7 \tan ^ { 8 } x + 7 \tan ^ { 6 } x - 3 \tan ^ { 4 } x - 3 \tan ^ { 2 } x , \quad \mathrm { I } _ { 1 } = \int _ { 0 } ^ { \pi / 4 } f ( x ) \mathrm { d } x$ and $\mathrm { I } _ { 2 } = \int _ { 0 } ^ { \pi / 4 } x f ( x ) \mathrm { d } x$. Then $7 \mathrm { I } _ { 1 } + 12 \mathrm { I } _ { 2 }$ is equal to:
(1) 2
(2) 1
(3) $2 \pi$
(4) $\pi$

If $\mathrm { I } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { \frac { 3 } { 2 } } x } { \sin ^ { \frac { 3 } { 2 } } x + \cos ^ { \frac { 3 } { 2 } } x } \mathrm {~d} x$, then $\int _ { 0 } ^ { 2\mathrm{I} } \frac { x \sin x \cos x } { \sin ^ { 4 } x + \cos ^ { 4 } x } \mathrm {~d} x$ equals :
(1) $\frac { \pi ^ { 2 } } { 12 }$
(2) $\frac { \pi ^ { 2 } } { 4 }$
(3) $\frac { \pi ^ { 2 } } { 16 }$
(4) $\frac { \pi ^ { 2 } } { 8 }$