Problem 39 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int \frac{1}{x^{2}} \sqrt{2-\frac{1}{x}} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{2}{3} (2 - \frac{1}{x})^{3/2} + C \).

Step 1: Substitution

Let's use the substitution method to simplify the integral. Set \( u = 2 - \frac{1}{x} \), which implies that \( du = \frac{1}{x^2} dx \). This simplifies our integral to \( \int \sqrt{u} \ du \).

Step 2: Integrate the New Expression

Now, integrate the expression \( \int \sqrt{u} \ du \), which is equal to \( \int u^{1/2} \ du \). This can be integrated using the power rule for integration: \( \int u^{n} \ du = \frac{u^{n+1}}{n+1} + C \). For \( n = 1/2 \), the integral becomes \( \frac{u^{3/2}}{3/2} + C \).

Step 3: Simplify the Integral Result

Simplifying the result of the integral, \( \frac{2}{3} u^{3/2} + C \).

Step 4: Substitute Back for x

Now, substitute back \( u = 2 - \frac{1}{x} \) to express the integral in terms of \( x \). The final result is \( \frac{2}{3} (2 - \frac{1}{x})^{3/2} + C \).

Key Concepts

Substitution Method

The Substitution Method is a powerful technique for simplifying integrals. In this method, we substitute a part of the integral with a variable, like changing clothes to better fit a situation. This makes the integration process more straightforward. For example, in the integral \( \int \frac{1}{x^2} \sqrt{2 - \frac{1}{x}} \, dx \), selecting an appropriate substitution can transform the problem and make it simpler to solve. Here, we chose \( u = 2 - \frac{1}{x} \) which simplifies the complexity of the square root term.After deciding on this substitution, we also find its derivative, \( du = \frac{1}{x^2} \, dx \). This derivative is crucial as it replaces \( \frac{1}{x^2} \, dx \) in the original integral, leading to a far simpler expression \( \int \sqrt{u} \, du \). In essence, substitution helps us step over obstacles by breaking them down into manageable tasks. It's like clearing a path through a dense forest, allowing us to focus on solving the much simpler form of the integral.

Power Rule for Integration

The Power Rule for Integration is akin to the arch-nemesis of differentiation's power rule. It provides a straightforward method to handle integrals of the form \( \int x^n \, dx \). The rule states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), assuming \( n eq -1 \). This rule simplifies the integration of polynomial terms. In our example, after applying the substitution method, we are left with an integral \( \int u^{1/2} \, du \). The term \( u^{1/2} \) fits perfectly into the power rule. By applying it: - The exponent \( n \) is \( 1/2 \). - Thus the integral becomes \( \frac{u^{3/2}}{3/2} + C \). This quickly converts a cumbersome looking square root into an expressible result that is easier to manipulate. The Power Rule for Integration is like using the right key for a specific lock, turning otherwise closed integrals into solvable expressions.

Definite and Indefinite Integrals

Integrals are best understood by comparing them to accumulative processes. We are often interested in two types: definite and indefinite integrals.

- **Indefinite Integrals**: These represent general forms of antiderivatives without specific bounds. They include an arbitrary constant \( C \) because integration is the reverse process of differentiation, which loses the constant during derivation. In our problem, we find \( \frac{2}{3} (2 - \frac{1}{x})^{3/2} + C \) as the indefinite integral.

- **Definite Integrals**: These calculate the accumulated value over a specific interval \([a, b]\). They do not include the constant \( C \) since the limits of integration set the bounds and fix the exact area. Unlike indefinite integrals, definite integrals provide a numerical result.

Converting an indefinite integral into a definite one involves adding limits, which is not present in our original problem. But understanding the difference prepares you for when you encounter bounds in your integrals.