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Question

Question: <img src="https://cdn.pureessence.tech/canvas_59.png?top_left_x=0&top_left_y=0&width=289&height=300"...

sgn (sin x) dx = …….. where sgn denotes signum of x

A

32\frac { 3 } { 2 } p

B

23\frac { 2 } { 3 }p2

C

32\frac { 3 } { 2 } p2

D

None of these

Answer

32\frac { 3 } { 2 } p2

Explanation

Solution

I = ∫0πx⋅sgn⁡(sin⁡x)dx+∫π2πx⋅sgn⁡(sin⁡x)dx\int _ { 0 } ^ { \pi } x \cdot \operatorname { sgn } ( \sin x ) d x + \int _ { \pi } ^ { 2 \pi } x \cdot \operatorname { sgn } ( \sin x ) d x

+∫2π3πxsgn⁡(sin⁡x)dx+ \int _ { 2 \pi } ^ { 3 \pi } x \operatorname { sgn } ( \sin x ) d x

= ∫0πx⋅1dx+∫π2πx(−1)dx+∫2π3πx⋅(1)dx\int _ { 0 } ^ { \pi } x \cdot 1 d x + \int _ { \pi } ^ { 2 \pi } x ( - 1 ) d x + \int _ { 2 \pi } ^ { 3 \pi } x \cdot ( 1 ) d x

= π22−12\frac { \pi ^ { 2 } } { 2 } - \frac { 1 } { 2 }(4p2 – p2) + 12\frac { 1 } { 2 }[9p2 – 4p2] = 3π22\frac { 3 \pi ^ { 2 } } { 2 }