$\int_{a}^{b} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a + b - x}} dx = . . . . . .$

$\int\limits_{-1}^{1} \frac{x^4}{1 + e^{x^7}} dx = $

$\int_0^1 \frac{\log _e(1+x)}{1+x^2} d x=$

If $\int_0^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \left(\frac{1}{2}\right)$,then $\int_0^{\frac{\pi}{2}} \log \sec x \, dx = $

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$ is equal to

$\int_{\frac{1}{3}}^3 \frac{1}{x} \sin \left(\frac{1}{x}-x\right) d x=$