Question

Find the area of evaluate $\int \sqrt{1-\sin \frac{\pi }{4}}dxx\in (\frac{\pi }{8},\frac{\pi }{4})$

Answer 1

$\int \sqrt{1-sin\frac{\pi }{4}}dx$ $x\in (\frac{\pi }{8},\frac{\pi }{4})$

$sin\frac{\pi }{4}=\frac{1}{\sqrt{2}}\Rightarrow \sqrt{1-\frac{1}{\sqrt{2}}}=\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}}}$

$=\int \sqrt{\frac{\sqrt{2}-1}{\sqrt{2}}}dx=\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}}}x.$