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Q.
The integral $\int^{\pi/2}_{0}\left|sin\,x-cos\,x\right|dx$ is equal to :
Integrals
Solution:
We have $cos\,x \ge sin\,x for 0\le x\le\frac{\pi}{4}$
and $sin\,x \ge cos\,x for \frac{\pi}{4}\le x \le\frac{\pi}{2}$
$\therefore \int^{\frac{\pi}{2}}_{0}\left|sin\,x-cos\,x\right|dx$
$=\int^{\frac{\pi}{4}}_{0}\left(cos\,x-sin\,x\right)dx+\int^{\frac{\pi}{2}}_{\frac{\pi}{4}}\left(sin\,x-cos\,x\right)dx$
$=\left[sin\,x+cos\,x\right]^{\frac{\pi}{4}}_{0}+\left[-cos\,x-sin\,x\right]^{\frac{\pi}{2}}_{\frac{\pi}{4}}$
$=\left[\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-0-1\right]-\left[0+1-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\right]$
$\sqrt{2}-1-1+\sqrt{2}=2\sqrt{2}-2$