Evaluate the definite integral $\int_{-1}^{1}(x+1) d x$.

$\int_0^{\pi /2} \frac{x + \sin x}{1 + \cos x} \, dx = $

Let $[x]$ denote the greatest integer $\leq x$. Consider the function $f(x) = \max \{x^2, 1 + [x]\}$. Then the value of the integral $\int_0^2 f(x) dx$ is:

Let $f(x) = \max \left\{3, x^2, \frac{1}{x^2}\right\}$ for $\frac{1}{2} \leq x \leq 2$. Then,the value of the integral $\int_{1/2}^2 f(x) dx$ is

If $[2,6]$ is divided into four intervals of equal length,then the approximate value of $\int_2^6 \frac{1}{x^2-x} dx$ using Simpson's rule is

Let $g(x) = \int_0^{|x|^{3/4}} t^{2/3} \sin \frac{1}{t} \, dt$,for all real $x$. Then,$\lim_{x \rightarrow 0} \frac{g(x)}{x}$ is equal to