91Ó°ÊÓ

Understanding the logarithmic identity is key when solving integrals like \(\int e^{2 \ln x} \, dx\). The identity we use here is \( e^{\ln a} = a \). This identity tells us that the function \( e^{\ln a} \) simplifies directly to \( a \) because the natural logarithm (\(\ln\)) and exponential function (\( e \)) are inverse functions.In the context of our exercise, \( e^{2 \ln x} \) can be rewritten. By using the properties of exponents and logarithms, we expand this expression:

• First, express it as \( (e^{\ln x})^2 \), meaning we are squaring \( e^{\ln x} \).
• Then, use the identity \( e^{\ln x} = x \) to further simplify it to \( x^2 \).

This simplification process is crucial as it transforms a complex-looking exponential function into a simple polynomial, \( x^2 \), which is much easier to integrate.

The power rule for integration is a fundamental tool when calculating antiderivatives of polynomial expressions. It's particularly useful after you've simplified an expression using logarithmic identities or other algebraic methods. The power rule states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \) is any real number and \( C \) is the constant of integration.In our exercise, once \( e^{2 \ln x} \) is simplified to \( x^2 \), we directly apply the power rule with \( n = 2 \):

• Integrate \( x^2 \) using the power rule.
• Thus, \( \int x^2 \, dx = \frac{x^{3}}{3} + C \).

This operation is straightforward because the power rule easily guides us to the antiderivative of a polynomial term like \( x^2 \). Remember, knowing when and how to apply the power rule is essential for integrating polynomials.

The substitution method is a powerful technique for integrating functions that are not immediately straightforward. However, in this exercise, after the application of the logarithmic identity, a direct substitution method was not necessary. Still, understanding its role in integration will benefit you in more complex situations.This method involves:

• Identifying a part of the integrand that can be replaced with a simpler variable \( u \).
• Then, differentiating \( u \) to find \( du \) and substitute it back into the integral.

For instance, while dealing with functions that involve products of expressions, a substitution might reduce the complexity, making the expression easier to integrate. Here, the starting integrand \( e^{2 \ln x} \) was simplified using logarithmic rules, rendering further substitution unnecessary. Nevertheless, being familiar with substitution is crucial as it allows you to tackle a wider range of integral problems.