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

If $y = e^{-mx}$ is a solution of the differential equation $\dfrac{d^2y}{dx^2} + 4\dfrac{dy}{dx} + 3y = 0$, then the values of $m$ are