Evaluate ${\int }_7^{23}\frac{dx}{x-2\sqrt{x+2}}$

For two integers $m,n,$ prove that

${\int }_0^{\pi }\sin mx\sin nxdx=\left\{\begin{array}{lll}0& \text{ when, }m\ne n& \\ \frac{\pi }{2}& \text{ when, }m=n\ne 0& \\ 0& \text{ when, }m=n=0& \end{array}\right.$

If $r=2(1-\cos \theta ),$ then show that ${\int }_0^{\pi }\sqrt{r^2+{\left(\frac{dr}{d\theta }\right)}^2}d\theta =8$.

Evaluate: 

$\sum _{n=1}^{1000}{\int }_{n-1}^n{e}^{x-\left[x\right]}dx$

Evaluate the following definite integrals by definition:

${\int }_a^{b}pdx$  [ where $p$ is a constant ]

Evaluate the following definite integrals by definition:

${\int }_0^{1}xdx$

Evaluate the following definite integrals by definition:

${\int }_1^2\left(2x+3\right)dx$

Evaluate the following definite integrals by definition:

${\int }_1^{4}3x^{2}dx$