Question

\( \int \left(e^{x\log_e 6}\right)e^x dx = \phi(x) + c \) then \( \phi(x) = \)

• A. \( 6^x e^x \)
• B. \( \frac{e^x}{\log 6e} \)
• C. \( \frac{6^x}{1+\log_e 6} \)
• D. \( \frac{(6e)^x}{1+\log_e 6} \)

Hint:

Use identity \( e^{x\ln a} = a^x \) to simplify exponential integrals quicklyThen apply standard \( \int a^x dx \) formula.

Answer 1

The Correct Option is D

Step 1: Simplify the given expression.
\[ e^{x\log_e 6} = 6^x \]
So the integral becomes:
\[ \int 6^x e^x dx \]

Step 2: Combine exponential terms.
\[ 6^x e^x = (6e)^x \]
Thus:
\[ \int (6e)^x dx \]

Step 3: Use standard formula.
\[ \int a^x dx = \frac{a^x}{\ln a} \]
So:
\[ \int (6e)^x dx = \frac{(6e)^x}{\ln(6e)} \]

Step 4: Simplify denominator.
\[ \ln(6e) = \ln 6 + \ln e = \ln 6 + 1 \]

Step 5: Write final expression.
\[ \phi(x) = \frac{(6e)^x}{1+\log_e 6} \]

Step 6: Match with options.
This matches option (D).

Step 7: Final conclusion.
\[ \boxed{\frac{(6e)^x}{1+\log_e 6}} \]