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Question: Evaluate : \(\int _ { 0 } ^ { \frac { 16 \pi } { 3 } } | \sin x |\) dx...

Evaluate : ∫016π3∣sin⁡x∣\int _ { 0 } ^ { \frac { 16 \pi } { 3 } } | \sin x | dx

A

214\frac { 21 } { 4 }

B

212\frac { 21 } { 2 }

C

112\frac { 11 } { 2 }

D

114\frac { 11 } { 4 }

Answer

212\frac { 21 } { 2 }

Explanation

Solution

Let I =

= ∫05π+π3∣sin⁡x∣\int _ { 0 } ^ { 5 \pi + \frac { \pi } { 3 } } | \sin x | dx

= dx + ∫5π5π+π3∣sin⁡x∣\int _ { 5 \pi } ^ { 5 \pi + \frac { \pi } { 3 } } | \sin x | dx

= 5 ∫0π∣sin⁡x∣\int _ { 0 } ^ { \pi } | \sin x | dx + dx

(By Prop. XIII & XV)

(Q | sin x| is periodic with period p)

= 5 dx + ∫0π/3sin⁡x\int _ { 0 } ^ { \pi / 3 } \sin x dx

= 5 (– cos x) ∣0π+(−cos⁡x)∣0π/3\left. \right| _ { 0 } ^ { \pi } + \left. ( - \cos x ) \right| _ { 0 } ^ { \pi / 3 }

= 5 (1 + 1) – (12−1)\left( \frac { 1 } { 2 } - 1 \right) = 212\frac { 21 } { 2 }