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UPSC NDA 2 2021 Mathematics Solved Paper

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Question : 84 of 120

Marks: +1, -0

What is

\int\limits_{0}^{\frac{\pi}{2}} e^{\ln(\cos x)}\,dx

equal to?

$-1$
0
1
2

Solution:

Concept:

The property

e^{\ln(f(x))} = f(x)

simplifies the integrand.

Explanation:

Let

I = \int_{0}^{\pi/2} e^{\ln(\cos x)} \, dx

.

Since

e^{\ln(\cos x)} = \cos x

, we have

I = \int_{0}^{\pi/2} \cos x \, dx

.

Evaluating:

\int \cos x \, dx = \sin x

, so

I = [\sin x]_{0}^{\pi/2}

.

Substitute limits:

\sin(\pi/2) - \sin(0) = 1 - 0 = 1

.

Thus the integral equals

1

.

Answer:

1

(option C).

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