$ 6\int_{0}^{\pi}|(\sin 3x+\sin 2x+\sin x)| dx $ is equal to ....

$\int_{0}^{2} ( |2x^2 - 3x| + [x - \frac{1}{2}] ) dx$,where $[ \cdot ]$ is the greatest integer function,is equal to:

Let $f: R \rightarrow R$ be a function defined as $f(x) = a \sin \left(\frac{\pi[x]}{2}\right) + [2-x]$,$a \in R$,where $[t]$ is the greatest integer less than or equal to $t$. If $\lim_{x \rightarrow -1} f(x)$ exists,then the value of $\int_{0}^{4} f(x) dx$ is equal to.

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

If $f(x)$ and $g(x)$ are inverse functions of each other such that $f(1) = 3$ and $f(3) = 1$,then $\int_{1}^{3} \left( g(x) + \frac{x}{f'(g(x))} \right) dx$ is equal to -

The value of the limit $\lim _{n \rightarrow \infty} \int _{0}^{1} x^{10} \sin (n x) d x$ equals