For \(x \gt 1, \int\displaystyle\frac{e^{7\log x}-e^{5\log x}}{e^{5\log x}-e^{4\log x}}\, dx\) equals.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{x^3}{3}+\frac{x^2}{2}+ C\): C is a constant of integration
Since \(e^{k\log x} = e^{\log x^k} = x^k\) for \(x \gt 0\), the integrand is a rational function of x:
\(\displaystyle \frac{x^7 - x^5}{x^5 - x^4} = \frac{x^5(x^2 - 1)}{x^4(x - 1)}\)
\(\displaystyle = \frac{x^5(x - 1)(x + 1)}{x^4(x - 1)} = x(x + 1)\)
(cancelling \(x - 1\) is allowed because \(x \gt 1\)). So
\(\displaystyle \int (x^2 + x)\,dx = \frac{x^3}{3} + \frac{x^2}{2} + C\)