If p(x) = (4e)²x}, then what is p(x) \, dx equal to?

Correct Answer is : p(x)2(1+2ln⁡2)+cp(x)/2(1+2 2) + c2(1+2ln2)p(x)+c

Correct Answer is : p(x)2(1+2ln⁡2)+cp(x)/2(1+2 2) + c2(1+2ln2)p(x)+c

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UPSC NDA I and II 2020 Math Solved Paper

Question : 63 of 120

Marks: +1, -0

p(x) = (4e)^{2x}

\int p(x) \, dx

equal to?

Solution:

Concept:

Integral of an exponential function:

\int a^{x} dx = \frac{a^{x}}{\ln a} + c

, where

a>0

and

a\neq 1

Explanation:

Given

p(x) = (4e)^{2x}

We need

I = \int p(x) \, dx = \int (4e)^{2x} dx

Substitute

2x = t

, so

2 dx = dt \Rightarrow dx = \frac{dt}{2}

Then

I = \frac{1}{2} \int (4e)^{t} dt = \frac{1}{2} \cdot \frac{(4e)^{t}}{\ln(4e)} + c

Simplify

\ln(4e) = \ln 4 + \ln e = \ln(2^2) + 1 = 2\ln 2 + 1 = 1 + 2\ln 2

Now

(4e)^{t} = (4e)^{2x} = p(x)

, so

I = \frac{1}{2} \cdot \frac{p(x)}{1+2\ln 2} + c = \frac{p(x)}{2(1+2\ln 2)} + c

Answer:

\frac{p(x)}{2(1+2\ln 2)} + c

(Option B).

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