Problem 3 - Applied Calculus

Question

Explain how you can tell if substitution can be used to find an antiderivative. $$\int \frac{\sqrt{\ln x}}{x} d x$$

Step-by-Step Solution

Answer

Check if substitution fits by identifying an inner function and its derivative in the integrand.

Step 1: Identify the Inner Function

In the given integral \( \int \frac{\sqrt{\ln x}}{x} \, dx \), the expression under the square root, \( \ln x \), can be considered as an inner function. The function \( \ln x \) suggests a potential substitution since it is part of a composite function (\( \sqrt{\ln x} \)).

Step 2: Compare with Derivative Form

The derivative of \( \ln x \) is \( \frac{1}{x} \), and we already have an \( \frac{1}{x} \) in the integrand. This is a strong indicator that substitution can be used, as the presence of \( \frac{1}{x} \, dx \) suggests the derivative of \( \ln x \) is part of the integration process.

Step 3: Define the Substitution

To apply substitution, set \( u = \ln x \). Then the derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = \frac{1}{x} \), or \( du = \frac{1}{x} \, dx \). This substitution transforms the integral into a simpler form.

Step 4: Transform the Integral

Replace \( \ln x \) with \( u \) and \( \frac{1}{x} \, dx \) with \( du \). The integral \( \int \frac{\sqrt{\ln x}}{x} \, dx \) becomes \( \int \sqrt{u} \, du \). This new integral is now integrable using basic rules.

Step 5: Integrate the New Integral

The integral of \( \sqrt{u} \) or \( u^{1/2} \) is \( \frac{u^{3/2}}{3/2} = \frac{2}{3}u^{3/2} + C \). This is the antiderivative of the transformed integral.

Step 6: Reverse the Substitution

After obtaining the antiderivative in terms of \( u \), substitute back \( u = \ln x \) to express the antiderivative in terms of the original variable \( x \). Therefore, the antiderivative is \( \frac{2}{3} (\ln x)^{3/2} + C \).

Key Concepts

Substitution Method

Understanding substitution is crucial for solving certain integration problems. When dealing with complicated integrals, identifying an inner function can be a game-changer. In the problem \( \int \frac{\sqrt{\ln x}}{x} \, dx \), the expression \( \ln x \) is an inner function. Recognizing this pattern suggests the substitution method is appropriate. The main idea is to transform the integral into something more manageable.

To do this, substitute \( u = \ln x \). The derivative \( \frac{du}{dx} = \frac{1}{x} \) indicates we have all necessary components. By rewriting the integrand in terms of \( u \), the integrand becomes simpler allowing for easier integration.

• Identify the inner function, often part of a composite function
• Check for the derivative of the inner function elsewhere in the integrand
• Transform the integral into a standard form

This aids in tackling problems that initially appear complex.

Antiderivatives

Antiderivatives reverse differentiation. They answer the question: "What function can I differentiate to get this integrand?" For example, given \( \int \sqrt{u} \, du \), we look for a function whose derivative returns \( \sqrt{u} \). As calculated, the antiderivative is \( \frac{2}{3}u^{3/2} + C \).

• Find the basic function family matching the integrand
• Adjust factors and powers to align with integration rules

Constants of integration \( C \) denote any constant value since differentiating a constant yields zero.
Solving includes
- Integrating with transformation
- Substituting back with initial variables

Reapplication of Original Variables

Once you have the antiderivative in terms of the substitution \( u \), remember to switch back to the original variable \( x \) to express the solution correctly.

Definite and Indefinite Integrals

Definite integrals provide numerical results, often representing areas under curves between specific bounds. Indefinite integrals, like in this exercise, imply a general form of the antiderivative plus a constant \( C \). Understanding this distinction is vital for correctly solving and interpreting the results.
For indefinite integrals:

• The focus is on determining a function and constant
• The constant is essential, representing all potential vertical shifts of the antiderivative

In our example, the integral \( \int \frac{\sqrt{\ln x}}{x} \, dx \) is indefinite. Hence, the solution \( \frac{2}{3} (\ln x)^{3/2} + C \) covers families of curves, showing all antiderivatives of the integrand.

On the other hand, if this were a definite integral, you'd calculate specific bounds to yield a numerical outcome, often related to physical quantities or phenomena.