Find ${\int }_0^{\pi /2} {\sin }^{7}xdx$

${\int }_0^{\frac{\pi }{2}} {\sin }^{n}xdx=\left\{\begin{array}{l}\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \frac{n-5}{n-4}\dots \cdot \frac{1}{2}\cdot \frac{\pi }{2}\text{ if }n\text{ iseven }\\ \frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \frac{n-5}{n-4}\dots \cdot \frac{2}{3}\text{ if }n\text{ is odd }\end{array}\right.$

${\int }_0^{\pi /2} {\sin }^{7}xdx=\frac{7-1}{7}\cdot \frac{7-3}{7-2}\cdot \frac{7-5}{7-4}\cdot 1$

$=\frac{6}{7}\cdot \frac{4}{5}\cdot \frac{2}{3}\cdot 1$

${\int }_0^{\pi /2} {\sin }^{7}xdx=\frac{16}{35}$