The value of $\int_0^\pi \dfrac{1}{1 + 2^{\cos x}}\,dx$ is _____

If $I_n = \int_1^e (\log_e x)^n\,dx$ where $n \in Z^+$ and $I_m + mI_{2026} = e$, then $m = $

The area of the region bounded by the lines $2x - y + 1 = 0,\ y = -1,\ y = 3$ and the y-axis is _____

Area enclosed by curve $y = 2x^2$ and lines $x \geq 1,\ y \leq 4$ is ..... sq. units