If $\int(x+2) \sqrt{x^2-x+2} \, dx = \frac{1}{3} f(x) + \frac{5}{8} g(x) + \frac{35}{16} h(x) + c$,then $f(-1) + g(-1) + h\left(\frac{1}{2}\right) = $

Let for $f(x)=7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$,$I_1 = \int_0^{\pi/4} f(x) \, dx$ and $I_2 = \int_0^{\pi/4} x f(x) \, dx$. Then $7 I_1 + 12 I_2$ is equal to:

For $k \in (1, \infty)$,$\int \frac{1}{1+k \cos x} d x=$

If $\frac{3 \pi}{2} < x < \frac{5 \pi}{2}$ and $\int(\sqrt{1-\sin x}+\sqrt{1+\sin x}) \, dx = f(x) + c$ where $c$ is the constant of integration,then $f\left(\frac{\pi}{3}\right) - f(0) =$

If $\int \frac{dx}{(x-1)^{3/2}(x-3)^{1/2}} = \sqrt{f(x)} + c$,then $f(-1) - f(0) =$

If $\int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[23]{3 x^{-24}+x^{-26}}\right) d x =-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^\gamma\right)^{\frac{\alpha+1}{\alpha}}+C, x>0,$ $(\alpha, \beta, \gamma \in Z)$,where $C$ is the constant of integration,then $\alpha+\beta+\gamma$ is equal to . . . . . . .