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

For $0 < x < \frac{\pi}{2}$,the integral $\int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} \ln(e^{\cos x}) \, d(\sin x)$ is equal to:

Let $I_{1}=\int_{0}^{n}[x] d x$ and $I_{2}=\int_{0}^{n}\{x\} d x,$ where $[x]$ and $\{x\}$ are the integral and fractional parts of $x$ respectively,and $n \in N-\{1\}.$ Then,$I_{1} / I_{2}$ is equal to

$ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+\cos x} = $

If $\int_0^b \frac{dx}{1+x^2} = \int_b^{\infty} \frac{dx}{1+x^2}$,then $b$ is equal to

$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin^2 x \, dx = $ . . . . . . .