$\int_{\frac{\pi}{4}}^{\frac{\pi}{4}} \log_e(\sin x + \cos x) \, dx$ is equal to

$ \int_{0}^{\frac{\pi}{2}} \frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x} d x $ is equal to

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

For a real number $x$,let $[x]$ denote the largest integer less than or equal to $x$ and $\{x\}=x-[x]$. The smallest possible integer value of $n$ for which $\int_1^n [x]\{x\} dx$ exceeds $2013$ is

$A$ minimum value of $\int_0^x t e^{t^2} d t$ is

If $f(x) = \begin{cases} \frac{6 x^2 + 1}{4 x^3 + 2 x + 3}, & 0 < x < 1 \\ x^2 + 1, & 1 \le x \le 2 \end{cases}$ then $\int_0^2 f(x) dx =$