Question

The integral I = $\int \frac{e^{5\log_e x} - e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}} dx$ is equal to

• A. $x + C$, where C is the constant of integration
• B. $\frac{x^2}{2} + C$, where C is the constant of integration
• C. $\frac{x^3}{3} + C$, where C is the constant of integration
• D. $\frac{x^4}{4} + C$, where C is the constant of integration

Hint:

Whenever you see expressions like \(e^{\ln f(x)}\), immediately simplify them to \(f(x)\). This is a common technique to make complex-looking integrals much simpler. Always look for opportunities to simplify by factoring.

Answer 1

The Correct Option is C

Step 1: Conceptual Understanding:
This problem necessitates simplifying the integrand using logarithmic properties prior to integration.
Step 2: Essential Formula/Method:
The crucial property to apply is \(e^{a \ln x} = e^{\ln(x^a)} = x^a\).
Post-simplification, the power rule for integration is used: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).
Step 3: Detailed Procedure:
Initially, simplify each term within the integrand by applying the property \(e^{a\ln x} = x^a\).


\(e^{5\log_e x} = x^5\)
\(e^{4\log_e x} = x^4\)
\(e^{3\log_e x} = x^3\)
\(e^{2\log_e x} = x^2\)
Substitute these simplified terms back into the integral:
\[ I = \int \frac{x^5 - x^4}{x^3 - x^2} dx \] Next, factorize both the numerator and the denominator:
\[ I = \int \frac{x^4(x - 1)}{x^2(x - 1)} dx \] Assuming \(x eq 1\) and \(x eq 0\), the common factor \((x-1)\) can be canceled:
\[ I = \int \frac{x^4}{x^2} dx \] \[ I = \int x^2 dx \] Subsequently, apply the power rule for integration:
\[ I = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C \] Step 4: Conclusive Result:
The integral evaluates to $\frac{x^3{3} + C$}.