Question

The value of ${\int }_{\pi }^{2\pi }\left[2\sin x\right]dx$ is equal to (where $[\cdot ]$ is the G.I.F.).

• A. $-\pi$
• B. $-2\pi$
• C. $-\frac{5\pi }{3}$
• D. $\frac{5\pi }{3}$

Answer 1

The correct option is C $-\frac{5\pi }{3}$

${\int }_{\pi }^{2\pi }\left[2\sin x\right]dx={\int }_{\pi }^{7\pi /6}\left[2\sin x\right]dx+{\int }_{7\pi /6}^{3\pi /2}\left[2\sin x\right]dx+{\int }_{3\pi /2}^{11\pi /6}\left[2\sin x\right]+{\int }_{11\pi /6}^{2\pi }\left[2\sin x\right]$

$={\int }_{\pi }^{7\pi /6}(-1)dx+{\int }_{7\pi /6}^{3\pi /2}(-2)dx+{\int }_{3\pi /2}^{11\pi /6}(-2)dx+{\int }_{11\pi /6}^{2\pi }(-1)dx$

$=[-x{]}_{\pi }^{7\pi /6}+[-2x{]}_{7\pi /6}^{3\pi /2}+[-2x{]}_{3\pi /2}^{11\pi /6}+[-x{]}_{11\pi /6}^{2\pi }$

$=\frac{-\pi }{6}+\frac{-2\pi }{3}+\frac{-2\pi }{3}+\frac{-\pi }{6}$

$=\frac{-5\pi }{3}$

The correct answer is $\frac{-5\pi }{3}$.