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UPSC NDA I 2021 Mathematics Solved Paper

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Question : 85 of 120

Marks: +1, -0

What is

\int e^{2\ln x + \ln x^2} dx

equal to?

$\frac{x^4}{4}+c$
$\frac{x^3}{3}+c$
$\frac{2x^5}{5}+c$
$\frac{x^5}{5}+c$

Solution:

Concept:

Use properties

\displaystyle e^{\ln a}=a

and

\displaystyle \ln a^m=m\ln a

.

Then integrate

\displaystyle \int x^n\,dx=\frac{x^{n+1}}{n+1}+C

.

Explanation:

Let

\displaystyle I=\int e^{2\ln x+\ln x^2}\,dx

.

Rewrite the exponent:

\displaystyle 2\ln x=\ln x^2

and

\displaystyle \ln x^2=2\ln x

.

Thus

\displaystyle 2\ln x+\ln x^2=\ln x^2+\ln x^2=2\ln x^2=\ln(x^2)^2=\ln x^4

.

So

\displaystyle I=\int e^{\ln x^4}\,dx=\int x^4\,dx

.

Apply the power rule:

\displaystyle \int x^4\,dx=\frac{x^5}{5}+C

.

Answer:

\displaystyle \frac{x^5}{5}+C

, which corresponds to option D.

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