If $f(x)$ satisfies the relation $f(x) = e^{x} + \int_{0}^{1} (y + xe^{x}) f(y) dy$, then $e + f(0)$ is equal to . . . . . . .

Let the function $f: R \rightarrow R$ be defined by
$f(t)=\begin{cases} (-1)^{n+1} 2, & \text{if } t=2n-1, n \in N \\ \frac{(2n+1-t)}{2} f(2n-1) + \frac{(t-(2n-1))}{2} f(2n+1), & \text{if } 2n-1 < t < 2n+1, n \in N \end{cases}$
Define $g(x) = \int_1^x f(t) dt, x \in (1, \infty)$. Let $\alpha$ denote the number of solutions of the equation $g(x) = 0$ in the interval $(1, 8]$ and $\beta = \lim_{x \rightarrow 1^+} \frac{g(x)}{x-1}$. Then the value of $\alpha + \beta$ is equal to.

Let $a, b, c$ be non-zero real numbers such that $\int_0^1 {(1 + \cos^8 x)(ax^2 + bx + c) \, dx} = \int_0^2 {(1 + \cos^8 x)(ax^2 + bx + c) \, dx}$. Then the quadratic equation $ax^2 + bx + c = 0$ has:

The greatest integer less than or equal to $\int_1^2 \log _2(x^3+1) dx + \int_1^{\log_2 9} (2^x-1)^{1/3} dx$ is . . . . .

The values of $\alpha$ which satisfy $\int_{\pi /2}^{\alpha} \sin x \, dx = \sin 2\alpha$,where $\alpha \in [0, 2\pi]$,are equal to:

Let $I_n = \int_{0}^{\frac{\pi}{4}} \tan^n x \, dx$. Then $\frac{1}{I_2 + I_4}, \frac{1}{I_3 + I_5}, \frac{1}{I_4 + I_6}, \dots$ are in: