The value of $\int_{-1 / 2}^{1 / 2} \cos ^{-1} x \, dx$ is

Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$,then $\int_0^a f(x) g(x) d x$ is equal to

$A$ function $f(x)$ satisfies $f(x) = f(\frac{c}{x})$ for some real number $c$ $(c > 1)$ and $\forall\, x > 0$. If $\int_{1}^{\sqrt{c}} \frac{f(x)}{x} dx = 3$,then the value of $\int_{1}^{c} \frac{f(x)}{x} dx$ is

Read the following mathematical statements carefully:
$I.$ $A$ differentiable function $f$ with maximum at $x = c$ $\implies f''(c) < 0$.
$II.$ Antiderivative of a periodic function is also a periodic function.
$III.$ If $f$ has a period $T$ then for any $a \in R$,$\int\limits_0^T {f(x)\,dx} = \int\limits_0^T {f(x + a)\,dx}$.
$IV.$ If $f(x)$ has a maxima at $x = c$,then $f$ is increasing in $(c - h, c)$ and decreasing in $(c, c + h)$ as $h \to 0$ for $h > 0$. Now indicate the correct alternative.

Let $A = \int\limits_0^1 \frac{e^t}{1 + t} \, dt$. Then $\int\limits_{a - 1}^a \frac{e^{-t}}{t - a - 1} \, dt$ has the value:

Let $J = \int_0^1 \frac{x}{1+x^8} dx$. Consider the following assertions:
$I$. $J > \frac{1}{4}$
$II$. $J < \frac{\pi}{8}$
Then,