Problem 43 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{\ln x}{x} d x \quad[\text { Hint: Let } u=\ln x .] $$

Step-by-Step Solution

Answer

The integral is \( \frac{(\ln x)^2}{2} + C \).

Step 1: Identify and Set Up the Substitution

The hint suggests using the substitution method. Let's let \( u = \ln x \). Then, we need to find the derivative of \( u \) with respect to \( x \).

Step 2: Differentiate \( u \) with respect to \( x \)

Differentiate the function \( u = \ln x \) to find \( \frac{du}{dx} \). This gives us \( \frac{du}{dx} = \frac{1}{x} \). Therefore, \( du = \frac{1}{x} dx \).

Step 3: Substitute and Simplify the Integral

With the substitution \( u = \ln x \) and \( du = \frac{1}{x} dx \), rewrite the integral. The integral becomes \( \int u \, du \).

Step 4: Integrate the Simplified Integral

The integral \( \int u \, du \) is a basic power integral. The antiderivative of \( u \) with respect to \( u \) is \( \frac{u^2}{2} \). So, \( \int u \, du = \frac{u^2}{2} + C \), where \( C \) is the constant of integration.

Step 5: Back Substitute \( u \)

Recall that \( u = \ln x \). Substitute back to obtain the integral in terms of \( x \). Thus, the integral becomes \( \frac{(\ln x)^2}{2} + C \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used frequently in integration. It's particularly useful when an integral includes a composition of functions. The goal is to simplify the integral by changing variables, typically by substituting part of the integrand with a single variable.
To effectively apply the substitution method, follow these steps:

• Identify an inner function in the integrand and assign it a new variable (like \( u \)).
• Find the differential of the new variable, \( du \), by differentiating your substituted equation with respect to the original variable, \( x \).
• Re-express the entire integral in terms of the new variable and its differential.
• Perform the integration with respect to the new variable.
• Finally, back substitute the original function into your result to return to the variable \( x \).

By making the integral simpler this way, substitution can transform a complex problem into an easy one.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a special type of logarithm that has some unique properties. It uses the mathematical constant \( e \), approximately equal to 2.718, as its base.
It is very prevalent in calculus and higher mathematics because of its nice properties related to growth rates and exponential functions.
Here are a few key things to remember:

• Natural logarithms are only defined for positive values of \( x \), meaning \( x > 0 \).
• The function \( \ln x \) is the inverse of the exponential function \( e^x \).
• The derivative of \( \ln x \) is \( \frac{1}{x} \), which makes it straightforward to use in calculations involving rates of change.

Understanding natural logarithms is crucial when dealing with integrals that include logarithmic expressions, as they frequently appear in various forms in integrands.

Integration Techniques

Integration techniques are a collection of strategies used to find antiderivatives or solve integrals.
While basic integration involves straightforward rules and formulas, often integrals require more advanced methods due to their complexity.
The substitution method is just one example of these techniques.
Here's a glimpse into some other common integration techniques:

• Integration by Parts: Useful for products of functions, it relies on the formula \( \int u \, dv = uv - \int v \, du \).
• Partial Fractions: Involves breaking down complex rational expressions into simpler fractions, which are easier to integrate.
• Trigonometric Substitution: Substitutes a trigonometric function in place of an expression to simplify integration, useful for integrals containing roots.
• Numerical Integration: Methods like Trapezoidal Rule or Simpson’s Rule approximate integrals when an analytical integration isn’t feasible.

The integration technique you choose depends on the form and complexity of the integral you're dealing with. Continually practicing these techniques will help you to flexibly handle a wide variety of problems.