Question

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

Short answer

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

Step-by-step solution

Recognize the Inner Function

Identify parts of the function in the integral that can be simplified using known algebraic identities. Notice that the expression \( e^{\ln x} \) can be simplified: since \( e^{\ln(x)} = x \). This is because \( e \) and \( \ln \) are inverse functions.

Simplify the Integral

Substitute the simplified form back into the integral. Therefore, the integral \( \int e^{\ln x} \frac{d x}{x} \) becomes \( \int x \frac{d x}{x} \), as the \( e^{\ln x} \) simplifies to \( x \). The equation now further simplifies to \( \int 1 \, dx \).

Integrate the Simplified Function

Perform the integration of \( \int 1 \, dx \). The integral of a constant is the constant multiplied by the variable of integration. Thus, we get \( x + C \), where \( C \) is the constant of integration.

Key concepts

Integration Techniques

Mastering integration techniques is essential in solving indefinite integrals. These techniques help in transforming complex integrals into simpler ones that are easy to evaluate. There are a variety of strategies to choose from, which often depend on the form of the function you are dealing with. Here are some common techniques:

• Substitution: Replacing part of the integral with a single variable to simplify the expression.
• Integration by Parts: This technique is based on the product rule for derivatives and is useful for integrating the product of functions.
• Trigonometric Integrals: Useful when dealing with trigonometric functions, sometimes involving identities.
• Partial Fractions: Breaking down complex rational expressions into simpler fractions.

In our exercise, we used a combination of recognizing function properties and simplifying with the substitution method. Techniques can often combine, tailored to fit the integral's particular form.

Substitution Method

The substitution method is a strategy commonly used to simplify the integration process. It works by substituting a part of the integrand with a new variable, simplifying the expression, and then integrating with respect to this new variable. Here's how it generally works:

• Identify a section of the integral that can be replaced with a simpler variable. This often involves spotting an inner function and its derivative.
• Replace this section with a new variable to transform the integral into an easier form.
• Integrate the new expression, and finally, replace the new variable with the original terms.

In our original exercise, recognizing that \( e^{\ln x} = x \) was crucial. Once the substitution was made, the integral became significantly easier to solve, essentially transforming it into a basic integration problem.

Algebraic Identities

Understanding algebraic identities is key to simplifying expressions within integrals. In integration, recognizing these identities can make complex integrals approachable.
In our exercise, the identity \( e^{\ln x} = x \) was pivotal. This stems from the relationship where exponential and logarithmic functions are inverse operations.
Some common algebraic identities include:

• Power Identities: Simplifications involving exponents like \( a^m \cdot a^n = a^{m+n} \).
• Logarithmic Identities: Such as \( \ln(a \cdot b) = \ln a + \ln b \).
• Trigonometric Identities: Useful in simplifying sequences involving sin, cos, and tan functions.

By recognizing these identities, you can effectively reduce complex integrals, often resulting in simpler forms that are straightforward to integrate. Applying these tools requires practice, but they form the backbone of efficient problem-solving in calculus.