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Question: \(\int_{a}^{b}\frac{|x|}{x}\)dx equals:...

∫ab∣x∣x\int_{a}^{b}\frac{|x|}{x}dx equals:

A

|a| – |b|

B

|b| – |a|

C

||a| – |b||

D

|b – a|

Answer

|b| – |a|

Explanation

Solution

Sol. Case -1

If a < b < 0, ∫ab∣x∣x\int_{a}^{b}\frac{|x|}{x}dx = – ∫abdx\int_{a}^{b}{dx}

= – (b –a) = |b| – |a|

(Q a < b < 0; |b| = – b and |a| = –a)

Case- 2

If a < 0 < b, ∫ab∣x∣xdx\int_{a}^{b}{\frac{|x|}{x}dx}= –∫a0dx\int_{a}^{0}{dx}+ ∫0bdx\int_{0}^{b}{dx}

= a + b = |b| – |a| (Q a < 0, |a| = –a)

Case- 3

Sol. If 0 < a < b, ∫ab∣x∣xdx\int_{a}^{b}{\frac{|x|}{x}dx}

= ∫abdx\int_{a}^{b}{dx}= b – a = |b| –|a|