$A$ value of $\alpha$ such that $\int_{\alpha}^{\alpha+1} \frac{dx}{(x+\alpha)(x+\alpha+1)} = \log_{e}\left(\frac{9}{8}\right)$ is

Number of values of $x$ satisfying the equation $\int_{-1}^{x} (8t^2 + \frac{28}{3}t + 4) dt = \frac{(\frac{3}{2})x + 1}{\log_{(x+1)} \sqrt{x+1}}$.

Evaluate the definite integral $\int_{2}^{3} \frac{1}{x} d x$.

Let $[t]$ denote the largest integer less than or equal to $t$. If $\int_0^3 \left( [x^2] + [\frac{x^2}{2}] \right) dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7}$,where $a, b, c \in \mathbb{Z}$,then $a + b + c$ is equal to:

The area of the region under the graph of $y = xe^{-ax}$ as $x$ varies from $0$ to $\infty$,where $'a'$ is a positive constant,is

The value of $I = \int_0^{\pi /2} \frac{(\sin x + \cos x)^2}{\sqrt{1 + \sin 2x}} dx$ is