Problem 340 - Calculus Volume 1

Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int e^{\ln x} \frac{d x}{x}$$

Step-by-Step Solution

Answer

The indefinite integral is \( x + C \).

Step 1: Simplify the Expression

First, recognize that the integrand can be simplified. Notice that \( e^{\ln x} = x \) because \( e^{\ln x} \) is just the inverse operation \( \ln x \) and raises \( x \) back to its base. Thus, the integral simplifies to \( \int \frac{x}{x} \, dx = \int 1 \, dx \).

Step 2: Integrate the Simplified Expression

With the integral now being \( \int 1 \, dx \), perform the integration which results in \( x + C \), where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

Integration is like reverse differentiation that helps find the original function given its derivative. To solve indefinite integrals, different integration techniques can be used. Some common techniques are:

• **Substitution Method:** Changing variables to simplify the integral.
• **Integration by Parts:** Derived from the product rule of differentiation.
• **Partial Fraction Decomposition:** Used to integrate rational functions by breaking them into simpler fractions.

In this exercise, the substitution method is employed. Recognizing and applying the right technique is crucial to simplifying and efficiently solving integrals.

Substitution Method

The substitution method is a powerful technique for solving integrals. It involves replacing a complicated part of the integral with a simpler variable. This method is effective when the integral has a function and its derivative present. Steps for using the substitution method:

• Identify a part of the integrand to substitute, often a function whose derivative also appears in the integral.
• Rewrite the integral in terms of the new variable.
• Integrate with respect to the new variable.
• Substitute back the original variable.

In the original exercise, you notice that substituting directly wasn't necessary because recognizing the simplification was enough to solve the integral. However, substitution often paves the way for simplifying complex expressions.

Simplification in Calculus

Simplification in calculus can transform a difficult integral into one that is easy to solve. Effective use of algebraic manipulation or recognizing equivalent expressions can dramatically alter an integral's complexity.In the exercise, we used a key property of logarithms and exponential functions: - Noticing that \(e^{\ln x} = x\) simplified the integrand elegantly.After the simplification, the integral \( \int \frac{x}{x} \, dx \) became \( \int 1 \, dx \).Simplifying before solving can save time and reduce errors. It lets you exploit underlying relationships between functions, showing that sometimes the path to the simplest solution is through recognizing fundamental identities.