Question

${\int }_{\mathrm{\pi }}^{16\mathrm{\pi }}\left|\sin x\right|dx=$

• A. $0$
• B. $32$
• C. $30$
• D. $28$

Answer 1

The correct option is C

$30$

Explanation for the correct option:

Finding the value for the given function:

Illustrating the graph of $y=\sin x$

In the curve of $\sin x$, the value from $0to\mathrm{\pi }$ becomes positive, from $\mathrm{\pi }to2\mathrm{\pi }$ it becomes negative,

again from $2\mathrm{\pi }to3\mathrm{\pi }$ the value becomes positive and from $3\mathrm{\pi }to4\mathrm{\pi }$ value becomes negative.

$$\begin{gathered} \Rightarrow {\int }_{\mathrm{\pi }}^{16\mathrm{\pi }}\left|\sin x\right|dx={\int }_{\mathrm{\pi }}^{2\mathrm{\pi }}-\left(\sin x\right)dx+{\int }_{2\mathrm{\pi }}^{3\mathrm{\pi }}\left(\sin x\right)dx+{\int }_{3\mathrm{\pi }}^{4\mathrm{\pi }}-\left(\sin x\right)dx+........+{\int }_{15\mathrm{\pi }}^{16\mathrm{\pi }}-\left(\sin x\right)dx \\ ={\left[\cos x\right]}_{\mathrm{\pi }}^{2\mathrm{\pi }}+{\left[-\cos x\right]}_{2\mathrm{\pi }}^{3\mathrm{\pi }}+{\left[\cos x\right]}_{3\mathrm{\pi }}^{4\mathrm{\pi }}+{\left[-\cos x\right]}_{4\mathrm{\pi }}^{5\mathrm{\pi }}.........+{\left[\cos x\right]}_{15\mathrm{\pi }}^{16\mathrm{\pi }} \\ =\left[1-\left(-1\right)\right]+\left[-\left\{\left(-1\right)-1\right\}\right]+\left[1-\left(-1\right)\right]+\left[-\left\{\left(-1\right)-1\right\}\right]........+\left[1-\left(-1\right)\right] \\ =2\times 15 \\ =30 \end{gathered}$$

Thus, option (C) is the correct answer