\( \int\displaystyle\frac{e^{7\log x}-e^{6\log x}}{e^{5\log x}-e^{4\log x}}\,dx,\ x\gt 0 \) is equal to :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{x^{3}}{3}+C \) : where \( C \) is an arbitrary constant
For \( x\gt 0 \), \( e^{n\log x}=x^{n} \), so the integrand becomes a ratio of powers of \( x \):
\( \displaystyle \frac{x^{7}-x^{6}}{x^{5}-x^{4}}=\frac{x^{6}(x-1)}{x^{4}(x-1)} \)
\( =x^{2} \)
Integrating,
\( \displaystyle \int x^{2}\,dx=\frac{x^{3}}{3}+C \)