$\int_{a-6}^{b-6} f(x+6) dx$ is equal to

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 = $

$\int_0^1 \frac{8 \log (1+x)}{1+x^2} \,d x=$

The value of the integral $\int \limits_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}$ is equal to

$\int_3^5(x-3)^3(5-x)^5 d x=$

If $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2x} \, dx = \alpha + \beta \sqrt{2} + \gamma \sqrt{3}$,where $\alpha, \beta$ and $\gamma$ are rational numbers,then $3\alpha + 4\beta - \gamma$ is equal to ..........