Mathematics - Definite Integration Question with Solution | TestHub

$P={\underset{\mathrm{n}\to \infty }{\lim }}_{}\prod _{r=1}^n{\left(\sin \left(\frac{r\pi }{4n}\right)\cos \left(\frac{r\pi }{4n}\right)\right)}^{\frac{1}{n}}$

$\ln P=\underset{n\to \infty }{\lim }\sum _{r=1}^n\ln \left(\frac{\sin r\pi }{n}\cdot \frac{\cos r\pi }{n}\right)$

$\ln P={\int }_0^1\ln \left(\sin \frac{\pi x}{4}\cdot \cos \frac{\pi x}{4}\right)dx$

$\ln P={\int }_0^1\ln \left(\sin \frac{\pi x}{2}\right)dx-\ln 2$

Put $\frac{\pi \mathrm{x}}{2}=\mathrm{t}$

$\therefore \ln P=\frac{2}{\pi }{\int }_0^{\frac{\pi }{2}}\ln (\sin t)dt-\ell n2$ $=-2\ln 2=\ln \frac{1}{4}$

$\therefore P=\frac{1}{4}\equiv \frac{a}{b}\Rightarrow a+b=5$