The integral \(I = \int\displaystyle\frac{e^{5\log_e x}-e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}}\,dx\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{x^3}{3}+C\), where C is the constant of integration
Since \(e^{k\log_e x} = e^{\log_e x^k} = x^k\) for x > 0, the exponentials reduce to powers of x:
\( \displaystyle I = \int \frac{x^5 - x^4}{x^3 - x^2}\,dx \)
Taking out common factors:
\( \displaystyle \frac{x^5 - x^4}{x^3 - x^2} = \frac{x^4(x - 1)}{x^2(x - 1)} = x^2 \)
So:
\( \displaystyle I = \int x^2\,dx = \frac{x^3}{3} + C \)
NCERT (2026-27 edition):
Integrals
