≤ft(e^{x _e 6}) e^x d x=(x)+c then (x)=

Correct Answer is : (6e)x1+log⁡e6(6 e)^x/1+ _e 61+loge6(6e)x

Correct Answer is : (6e)x1+log⁡e6(6 e)^x/1+ _e 61+loge6(6e)x

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COMEDK 2025 Evening Solved Paper

Question : 148 of 180

Marks: +1, -0

\int\left(e^{x \log _e 6}\right) e^x d x=\phi(x)+c

\phi(x)=

Solution:

Concept:

Simplify the integrand using exponent and logarithm rules, then apply the standard integral of an exponential function.

Explanation:

First, note

e^{x \log_e 6} = 6^x

because

\log_e 6 = \ln 6

Then

e^{x \ln 6} \cdot e^x = 6^x \cdot e^x = (6e)^x

Integrate:

\int (6e)^x dx = \frac{(6e)^x}{\ln(6e)} + C

Since

\ln(6e) = \ln 6 + \ln e = \ln 6 + 1

, we get

\phi(x) = \frac{(6e)^x}{1 + \ln 6}

Note

\ln 6 = \log_e 6

, so the denominator is

1 + \log_e 6

. This corresponds to option D.

Answer:

\phi(x) = \frac{(6e)^x}{1 + \log_e 6}

, i.e., option D.

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