Question

$ \int \frac{e^{10 \log x} - e^{8 \log x}}{e^{6 \log x} - e^{5 \log x}} \, dx$ is equal to :

• A. $x + C$
• B. $\frac{x^2}{2} + C$
• C. $\frac{x^4}{4} + C$
• D. $\frac{x^3}{3} + C$

Hint:

When simplifying expressions involving logarithmic terms, convert the terms to powers of $x$ using properties of logarithms.

Answer 1

The Correct Option is B

The given expression simplifies to: \[ \frac{e^{10 \log x} - e^{8 \log x}}{e^{6 \log x} - e^{5 \log x}} = \frac{x^{10} - x^8}{x^6 - x^5} \] Simplifying further: \[ \frac{x^8(x^2 - 1)}{x^5(x - 1)} = x^3 \] Thus, the integral is: \[ \int x^3 \, dx = \frac{x^4}{4} + C \] The correct answer is $(B)$.