What is e^{x (a)} \, dx equal to?

Correct Answer is : axln⁡(a)+ca^x/≤ft(a) + cln(a)ax+c

Correct Answer is : axln⁡(a)+ca^x/≤ft(a) + cln(a)ax+c

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UPSC NDA I 2019 Mathematics Solved Paper

Question : 94 of 120

Marks: +1, -0

What is

\int e^{x \ln(a)} \, dx

equal to?

Solution:

Concept:

Use the identity

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

to simplify the integrand.

Explanation:

Step 1: Rewrite the exponent:

x \ln a = \ln (a^x)

Therefore,

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

Step 2: The integral becomes

\displaystyle \int a^x \, dx

Step 3: Apply the standard formula

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

, where

C

is the constant of integration.

Step 4: Hence

\int e^{x \ln a} \, dx = \frac{a^x}{\ln a} + C

Answer:

\displaystyle \frac{a^x}{\ln a} + c

(Option A)

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