Test Index

UPSC NDA I 2017 Mathematics Solved Paper

Question : 119 of 120

Marks: +1, -0

What is

∫_{e^{-1}}^{e^{2}} | \frac{lnx}{x} | dx

equal to?

Solution:

Concept:

The absolute value makes the integrand positive. The sign of

\ln x

changes at

x=1

, so the integral must be split at that point.

Explanation:

For

x

between

e^{-1}

and

1

\ln x < 0

, so

\left|\frac{\ln x}{x}\right| = -\frac{\ln x}{x}

For

x

between

1

and

e^{2}

\ln x > 0

, so

\left|\frac{\ln x}{x}\right| = \frac{\ln x}{x}

Hence, split the integral:

\int_{e^{-1}}^{e^{2}} \left|\frac{\ln x}{x}\right| dx = \int_{e^{-1}}^{1} -\frac{\ln x}{x} dx + \int_{1}^{e^{2}} \frac{\ln x}{x} dx.

Use substitution

t = \ln x

, so

\frac{dx}{x} = dt

Limits:

x=e^{-1} \to t=-1

;

x=1 \to t=0

;

x=e^{2} \to t=2

The integral becomes:

\int_{-1}^{0} -t \, dt + \int_{0}^{2} t \, dt = \left[-\frac{t^{2}}{2}\right]_{-1}^{0} + \left[\frac{t^{2}}{2}\right]_{0}^{2}.

Evaluate: first part =

0 - \left(-\frac{1}{2}\right) = \frac{1}{2}

; second part =

2 - 0 = 2

Sum =

\frac{1}{2} + 2 = \frac{5}{2}

Answer:

\frac{5}{2}

(Option B).

Go to Question:

Prev Question

Next Question