Problem 47 - Calculus and its applications

Question

Evaluate. $$ \int \sqrt{x} \ln x d x $$

Step-by-Step Solution

Answer

\( \int \sqrt{x} \ln x \, dx = \frac{2}{3} x^{3/2} \ln x - \frac{4}{9} x^{3/2} + C \).

Step 1: Identify the Integration Method

To evaluate the integral \( \int \sqrt{x} \ln x \, dx \), notice that it involves the product of two functions, \( \sqrt{x} \) and \( \ln x \). Hence, we will use integration by parts, which is given by the formula \( \int u \, dv = uv - \int v \, du \).

Step 2: Choose \( u \) and \( dv \)

Select \( u = \ln x \) and \( dv = \sqrt{x} \, dx = x^{1/2} \, dx \). The reason for this choice is that differentiating \( \ln x \) simplifies the integral, while integrating \( x^{1/2} \) is straightforward.

Step 3: Differentiate \( u \) and Integrate \( dv \)

Differentiate \( u = \ln x \), which gives \( du = \frac{1}{x} \, dx \). Integrate \( dv = x^{1/2} \, dx \), resulting in \( v = \frac{2}{3}x^{3/2} \) by applying the power rule.

Step 4: Apply the Integration by Parts Formula

Substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula: \[ \int \sqrt{x} \ln x \, dx = \ln x \cdot \frac{2}{3}x^{3/2} - \int \frac{2}{3}x^{3/2} \cdot \frac{1}{x} \, dx. \]

Step 5: Simplify the Second Integral

Simplify \( \int \frac{2}{3}x^{3/2} \cdot \frac{1}{x} \, dx \) to \( \int \frac{2}{3}x^{1/2} \, dx \), using the property \( x^{3/2 - 1} = x^{1/2} \). This reduces the problem to a basic power rule integration.

Step 6: Evaluate the Remaining Integral

Evaluate \( \int \frac{2}{3}x^{1/2} \, dx \) by applying the power rule: \( \int x^{1/2} \, dx = \frac{2}{3}x^{3/2} \). Thus, \[ \int \frac{2}{3}x^{1/2} \, dx = \frac{2}{3} \cdot \frac{2}{3} x^{3/2} = \frac{4}{9}x^{3/2}. \]

Step 7: Assemble the Final Result

Replace the simplified integral back into the equation from Step 4: \[ \int \sqrt{x} \ln x \, dx = \frac{2}{3} x^{3/2} \ln x - \frac{4}{9} x^{3/2} + C, \] where \( C \) is the constant of integration.

Key Concepts

Integration by Parts

Integration by parts is a technique derived from the product rule for differentiation. It helps us integrate the product of two functions by transforming it into an expression involving the integration of one function. This transformation makes it possible to simplify complex integrals.
For example, when faced with an integral like \( \int \sqrt{x} \ln x \, dx \), we can utilize the formula:

• \( \int u \, dv = uv - \int v \, du \)

This formula requires choosing which part of the integral becomes \( u \) and which becomes \( dv \). A good strategy is choosing \( u \) as the function that simplifies upon differentiation, while \( dv \) should be easily integrable. In our example, setting \( u = \ln x \) and \( dv = \sqrt{x} \, dx = x^{1/2} \, dx \) works well, because the derivative of \( \ln x \) is straightforward, and the integration of \( x^{1/2} \) is simple using the power rule.

Indefinite Integrals

An indefinite integral is essentially the reverse process of differentiation. It represents a family of functions, and does not include specific limits of integration. Instead, it involves adding a constant of integration, denoted as \( C \). This accounts for the fact that many functions differ only by a constant when their derivatives are equivalent.
In the context of our integral \( \int \sqrt{x} \ln x \, dx \), we aim to find a function whose derivative yields the integrand, \( \sqrt{x} \ln x \). After employing integration by parts and simplifying, we derive:

• \( \frac{2}{3} x^{3/2} \ln x - \frac{4}{9} x^{3/2} + C \)

The \( +C \) is crucial as it represents an infinite number of possible solutions, translating the broad nature of antiderivatives.

Power Rule in Integration

The power rule in integration is a fundamental technique used to integrate functions of the form \( x^n \), where \( n \) is not equal to \(-1\). It states that the integral of \( x^n \) with respect to \( x \) is

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) if \( n eq -1 \)

In the exercise, the power rule is employed when integrating \( x^{1/2} \, dx \), which simplifies to \( \frac{2}{3} x^{3/2} \). This application is crucial, for it reduces integrals to simpler terms and helps evaluate the remaining portions of the integral after applying integration by parts. Breaking down complex expressions into standard power forms allows easier computation and systematic handling of antiderivatives. Understanding and mastering the power rule is vital for solving a wide array of calculus problems efficiently.