Question

Find the value of the definite integral $I={\int }_0^{\pi }|\sqrt{2}\sin x+2\cos x|dx$.

Answer 1

Consider the given integral.

$I={\int }_0^{\pi }|\sqrt{2}\sin x+2\cos x|dx$

Solve the definite integration.

$I={\int }_0^{\pi }|\sqrt{2}\sin x+2\cos x|dx$

$={\int }_0^{\frac{\pi }{2}}(\sqrt{2}\sin x+2\cos x)dx+{\int }_{\frac{\pi }{2}}^{\pi }(-\sqrt{2}\sin x-2\cos x)dx$

$={\int }_0^{\frac{\pi }{2}}(\sqrt{2}\sin x+2\cos x)dx+{\int }_{\frac{\pi }{2}}^{\pi }(\sqrt{2}\sin x+2\cos x)dx$

$={(-\sqrt{2}\cos x+2\sin x)}_0^{\frac{\pi }{2}}+{(-\sqrt{2}\cos x+2\sin x)}_{\frac{\pi }{2}}^{\pi }$

$=((-\sqrt{2}\cos \frac{\pi }{2}+2\sin \frac{\pi }{2})-(-\sqrt{2}\cos 0+2\sin 0))+((-\sqrt{2}\cos \frac{\pi }{2}+2\sin \frac{\pi }{2})-(-\sqrt{2}\cos \pi +2\sin \pi ))$

$=6.82$

Hence, this is the required result.