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UPSC NDA 2 2022 Mathematics Solved Paper

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Question : 74 of 120

Marks: +1, -0

What is ∫(xx)2 (1 + ln x)dx equal to ?

x2x + c
$\frac{1}{2} x^{2x} + c$
2x2x + c
$\frac{1}{2}x^{x} + c$

Solution:

Concept:

Use substitution

x^{2x} = u

to simplify the integral.

Explanation:

Let

u = x^{2x}

.

Take natural logarithm:

\ln u = 2x \ln x

.

Differentiate both sides:

\frac{1}{u} du = (2 \ln x + 2) dx = 2(1 + \ln x) dx

.

Thus,

(1 + \ln x) dx = \frac{1}{2u} du

.

The integral becomes

\int x^{2x} (1 + \ln x) dx = \int u \cdot \frac{1}{2u} du = \frac{1}{2} \int du

.

Integrate:

\frac{1}{2} u + c = \frac{1}{2} x^{2x} + c

.

Answer:

\frac{1}{2} x^{2x} + c

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