What is _{-1}^0 h(x) d x equal to ?

Correct Answer is : −32-3/2−23

Correct Answer is : −32-3/2−23

Test Index

UPSC NDA I 2024 Mathematics Paper

Question Numbers: 93-94

Consider the following for the items that follow :

Let f(x) = | x - 1 | , g(x) = [x] and h(x) = f(x)g(x) where [.] is greatest integer function.

Question : 93 of 120

Marks: +1, -0

What is

\int_{-1}^0 h(x) d x

equal to ?

Solution:

Concept:

We need to evaluate

\int_{-1}^{0} h(x) dx

where

h(x)=f(x)g(x)

with

f(x)=|x-1|

and

g(x)=[x]

(greatest integer function).

Explanation:

For

x

[-1,0]

, the greatest integer

[x] = -1

Since

x < 1

f(x) = |x-1| = -(x-1) = 1-x

Thus

h(x) = (1-x)(-1) = x-1

The integral becomes

\int_{-1}^{0} (x-1) dx

Evaluate:

\left[\frac{x^{2}}{2} - x\right]_{-1}^{0} = (0-0) - \left(\frac{(-1)^{2}}{2} - (-1)\right) = -\left(\frac{1}{2}+1\right) = -\frac{3}{2}

Answer:

-\frac{3}{2}

(Option A).

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