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UPSC NDA 2 2021 Mathematics Solved Paper

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Question : 96 of 120

Marks: +1, -0

If

f(x) = 2^{x}

\int\limits_{2}^{10} \frac{f'(x)}{f(x)} \, dx

equal to?

4 ln 2
ln 4
ln 5
8 ln 2

Solution:

Concept:

Use the formula

\frac{d}{dx} a^{x} = a^{x} \ln a

and the property

\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C

.

Explanation:

We have

f(x) = 2^{x}

.

Then

f'(x) = 2^{x} \ln 2

.

Therefore,

\frac{f'(x)}{f(x)} = \frac{2^{x} \ln 2}{2^{x}} = \ln 2

.

Now integrate:

\int_{2}^{10} \frac{f'(x)}{f(x)} dx = \int_{2}^{10} \ln 2 \, dx

.

Since

\ln 2

is a constant,

\int_{2}^{10} \ln 2 \, dx = \ln 2 \, [x]_{2}^{10} = \ln 2 \, (10 - 2) = 8 \ln 2

.

Answer:

8 \ln 2

(Option D).

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