Let $f(x)$ be a continuous periodic function with period $T$. Let $I = \int_{a}^{a+T} f(x) \, dx$. Then

If $I_n = \int_0^{\pi / 4} \tan^n x \, dx$,then $I_2+I_4, I_3+I_5, I_4+I_6, \ldots$ are in

$\int_0^\pi \frac{x \tan x}{\sec x+\cos x} \,d x=$

$\int_0^\pi (\sin^5 x \cos^3 x + \sin^4 x \cos^4 x + \sin^3 x \cos^4 x) dx =$

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 the function $F$ be defined as $F(x) = \int_{1}^{x} \frac{e^{t}}{t} dt, x > 0$. Then the value of the integral $\int_{1}^{x} \frac{e^{t}}{t+a} dt$,where $a > 0$,is