What is ₁/3}¹ f(x)\,dx equal to ? [NDA I 2016]

Correct Answer is : 377237/727237

Correct Answer is : 377237/727237

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UPSC NDA I 2016 Mathematics Solved Paper

For th next two (2) items that follow :

Consider the function

f(x)=x g(x)

and

g(x)=\frac{1}{[x]}

Where

[\cdot]

is the greatest integer function

[NDA I 2016]

Question : 17 of 120

Marks: +1, -0

What is

\int\limits_{\frac{1}{3}}^{1} f(x)\,dx

Solution:

\int\limits_{\frac{1}{3}}^{1} f(x)\,dx = \int\limits_{\frac{1}{3}}^{\frac{1}{2}} f(x)\,dx + \int\limits_{\frac{1}{2}}^{1} f(x)\,dx

......(1)

f(x)=x g(x)

g\left(\frac{1}{3}\right)=3g\left(\frac{1}{2}\right)=2g(1)=1

The value of

g(x)

in value

\left(\frac{1}{2},\frac{1}{3}\right)

will be 2 and in range

\left(\frac{1}{2},1\right)

it will be 1

From

(1)

\int\limits_{\frac{1}{3}}^{1} f(x)\,dx = \int\limits_{\frac{1}{3}}^{\frac{1}{2}} x g(x)\,dx + \int\limits_{\frac{1}{2}}^{1} x g(x)\,dx

= \int\limits_{\frac{1}{3}}^{\frac{1}{2}} x \cdot 2\,dx + \int\limits_{\frac{1}{2}}^{1} x \times 1\,dx

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

= \left[ \frac{1}{4} - \frac{1}{9} \right] + \frac{1}{2} \left[ 1 - \frac{1}{4} \right]

= \frac{5}{36} + \frac{3}{8} = \frac{37}{72}

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