Which of the following statements is incorrect for the function $g(\alpha)$ for $\alpha \in R$ such that

$g(\alpha)=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin ^{\alpha} x}{\cos ^{\alpha} x+\sin ^{\alpha} x} d x$

If $\alpha \in (2 , 3) $ then number of solution of the equation $\int\limits_0^\alpha  {}  \cos (x + \alpha^2)\, dx = \sin \,\alpha$ is :

Let $\operatorname{Max} \limits _{0 \leq x \leq 2}\left\{\frac{9-x^{2}}{5-x}\right\}=\alpha$ and $\operatorname{Min} \limits _ {0 \leq x \leq 2}\left\{\frac{9-x^{2}}{5-x}\right\}=\beta$

If $\int\limits_{\beta-\frac{8}{3}}^{2 a-1} \operatorname{Max}\left\{\frac{9- x ^{2}}{5- x }, x \right\} dx =\alpha_{1}+\alpha_{2} \log _{e}\left(\frac{8}{15}\right)$ then $\alpha_{1}+\alpha_{2}$ is equal to

Let for $x \in R , S_0( x )= x$,$S _{ k }( x )= C _{ k } x + k \int _0^{ x } S _{ k -1}(t) d t$, where $C _0=1, C _{ k }=1-\int_0^1 S _{ k -1}( x ) dx , k =1,2,3 \ldots$. Then $S _2(3)+6 C _3$ is equal to $...........$.

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1, \mathrm{f}(1))$ and $(3, \mathrm{f}(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x$-axis. If $27 \int_1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3} \quad$ where $\alpha, \quad \beta$ are integers, then the value of $\alpha+\beta$ equals