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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Definite Integration

NDA 2015 · Mathematics · Definite Integration

If 0 < a < b , then ∫_a^b (|x|)/(x) d x is equal to

If 0<a<b0 < a < b, then ∫ab∣x∣xdx\int_a^b \frac{|x|}{x} d x is equal to

Options

  1. A∣b∣−∣a∣|\mathrm{b}|-|\mathrm{a}|
  2. B∣a∣−∣b∣|\mathrm{a}|-|\mathrm{b}|
  3. C∣b∣∣a∣\frac{|\mathrm{b}|}{|\mathrm{a}|}
  4. D0

Correct answer

A. ∣b∣−∣a∣|\mathrm{b}|-|\mathrm{a}|

Step-by-step solution

∫ab∣x∣xdxwhen x≥0⇒∫abxxdx⇒∫ab(1)dx=[x]ab=∣b∣−∣a∣when x<0; as 0<a<b;x will not lie between a and bso ba∣x∣xdx=0 for x<0a\begin{aligned} & \int_a^b \frac{|x|}{x} d x \\ & \text{when } x \geq 0 \\ & \Rightarrow \int_a^b \frac{x}{x} d x \\ & \Rightarrow \int_a^b(1) d x \\ & =[x]_a^b \\ & =|b|-|a| \\ & \text{when } x < 0 ; \text{ as } 0 < a < b ; x \text{ will not lie between a and } b \\ & \text{so } \\ & \frac{b}{a} \frac{|x|}{x} d x=0 \text{ for } x < 0 \\ & a\end{aligned}
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