Question

The value of $\underset{0}{\overset{2\pi }{\int }}\left|\cos x-\sin x\right|dx$ is equal to

• A. $2\sqrt{2}$
• B. 2
• C. 4
• D. $4\sqrt{2}$

Answer 1

The correct option is D $4\sqrt{2}$

$\underset{0}{\overset{2\pi }{\int }}\left|cosx-sinx\right|dx=\underset{0}{\overset{\frac{\pi }{4}}{\int }}(cosx-sinx)+\underset{\frac{\pi }{4}}{\overset{\frac{5\pi }{4}}{\int }}(sinx-cosx)+\underset{\frac{5\pi }{4}}{\overset{2\pi }{\int }}(cosx-sinx)$

$=(sinx+cosx{)}_0^{\frac{\pi }{4}}+(-cosx-sinx{)}_{\frac{\pi }{4}}^{\frac{5\pi }{4}}+(sinx+cosx{)}_{\frac{5\pi }{4}}^{2\pi }$

$=\sqrt{2}-1+2\sqrt{2}+1+\sqrt{2}$

$=4\sqrt{2}$