Question

Evaluate \( \int \frac{e^{6 \log x} - e^{5 \log x}}{e^{4 \log x} - e^{3 \log x}} \, dx \):

• A. \( e^x + C \)
• B. \( \frac{x^2}{2} + C \)
• C. \( x + C \)
• D. \( \frac{x^3}{3} + C \)
• E. \( x e^x + C \)

Hint:

Rewriting exponential terms using logarithmic properties can simplify the integral greatly.

Answer 1

The Correct Option is D

We start by rewriting the exponentials in terms of powers of \( x \): \[ e^{6 \log x} = x^6, \quad e^{5 \log x} = x^5, \quad e^{4 \log x} = x^4, \quad e^{3 \log x} = x^3 \] Replacing these expressions in the integral, we obtain: \[ \int \frac{x^6 - x^5}{x^4 - x^3} \, dx \] Factoring \( x^3 \) from both the numerator and denominator: \[ \int \frac{x^3(x^3 - x^2)}{x^3(x - 1)} \, dx = \int \frac{x^3 - x^2}{x - 1} \, dx \] Solving the integral results in: \[ \frac{x^3}{3} + C \] Thus, the final answer is \( \frac{x^3}{3} + C \).