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UPSC NDA II 2015 Mathematics Paper

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Question : 63 of 120

Marks: +1, -0

If 0 < a < b, then

\int\nolimits_{\mathrm{a}}^{\mathrm{b}} \frac{\left|x\right|}{x}\,\mathrm{d}x

is equal to

| b | - | a |
| a | - | b |
$\frac{{\left | {\rm{b}} \right | }}{{\left | {\rm{a}} \right | }}$
0

Solution:

Concept:

For

x>0

,

|x| = x

, so the integrand simplifies to

1

.

Explanation:

Given

0 < a < b

, the interval

[a,b]

contains only positive values of

x

.

Hence,

|x| = x

for all

x

in

[a,b]

.

The integrand becomes

\frac{|x|}{x} = \frac{x}{x} = 1

.

Thus,

\int_a^b \frac{|x|}{x} dx = \int_a^b 1 \, dx = [x]_a^b = b - a

.

Since

a>0

and

b>0

, we have

b - a = |b| - |a|

.

Answer:

Option A:

|b| - |a|

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