Problem 57 - Calculus Early Transcendentals

Question

Different methods a. Evaluate \(\int \frac{x^{2}}{x+1} d x\) using the substitution \(u=x+1\) b. Evaluate \(\int \frac{x^{2}}{x+1} d x\) after first performing long division on the integrand. c. Reconcile the results in parts (a) and (b).

Step-by-Step Solution

Answer

Question: Evaluate the integral \(\int \frac{x^{2}}{x+1} dx\) using substitution and long division and reconcile the results. Solution: Using substitution, we get the result: $$\frac{1}{2}(x+1)^2 - 2(x+1) + \ln|x+1| + C$$ Using long division, we get the result: $$\frac{1}{2}x^2 - x + \ln|x+1| + C$$ By comparing and simplifying the first expression, we can reconcile the results, which shows both expressions are equal: $$\frac{1}{2}x^2 - x + \ln|x+1| + C$$

Step 1: Identify the substitution parameter

Choose \(u = x+1\), so \(x = u-1\). We need to find the derivative of \(u\) with respect to \(x\): \(du = dx\).

Step 2: Rewrite the integral in terms of u

Substitute the parameter and its derivative into the integral and simplify: $$\int \frac{(u-1)^{2}}{u} du$$

Step 3: Expand and simplify the integrand

Expand \((u-1)^2\) and simplify the integrand: $$\int \frac{u^2 - 2u + 1}{u} du = \int (u - 2 + \frac{1}{u}) du$$

Step 4: Perform the integration

Now, integrate the resulting expression with respect to \(u\): $$\int (u - 2 + \frac{1}{u}) du = \frac{1}{2}u^2 - 2u + \ln|u| + C$$

Step 5: Convert the result back to x

Replace \(u\) with the original substitution: \(x+1\): $$\frac{1}{2}(x+1)^2 - 2(x+1) + \ln|x+1| + C$$ Method 2: Long Division

Step 1: Perform the long division on the integrand

Divide \(x^2\) by \(x+1\) and find the remainder: $$ x^2 = (x+1)(x-1) + 1 $$

Step 2: Rewrite the integral using the result of the long division

Express the integrand as the sum of its division and remainder: $$\int \frac{x^2}{x+1} dx = \int (x - 1 + \frac{1}{x+1}) dx$$

Step 3: Perform the integration

Integrate the resulting expression with respect to \(x\): $$\int (x - 1 + \frac{1}{x+1}) dx = \frac{1}{2}x^2 - x + \ln|x+1| + C$$ Reconciling the results We have found two expressions for the integral: 1. $$\frac{1}{2}(x+1)^2 - 2(x+1) + \ln|x+1| + C$$ 2. $$\frac{1}{2}x^2 - x + \ln|x+1| + C$$ To reconcile, notice that \((x+1)^2 = x^2 + 2x + 1\) and \(2(x+1) = 2x + 2\). Replacing in the first expression, we get: $$\frac{1}{2}(x^2 + 2x + 1) - (2x + 2) + \ln|x+1| + C$$ Simplifying the expression, we get: $$\frac{1}{2}x^2 - x + \ln|x+1| + C$$ Both expressions are equal, thus the results from both methods are reconciled.

Key Concepts

U-Substitution in Integration

The technique of u-substitution is a method for simplifying integrals by transforming the original variable into a new one. It's the reverse process of the chain rule in differentiation and is especially useful when dealing with composite functions.

Here's how to implement u-substitution effectively:

• Identify a function within the integral that, when differentiated, appears elsewhere in the integral.
• Define this function as 'u' and write down 'du', its differential.
• Express the original integral in terms of 'u' and 'du', replace all x terms.
• Perform the integration with respect to 'u'.
• Substitute back the expression for 'u' in terms of x into the final result.

Following these steps transforms the initial complex integral into a more manageable form. For example, given \(\int \frac{x^{2}}{x+1} dx\), using \(u = x + 1\) leads to a simpler expression \(\int (u - 2 + \frac{1}{u}) du\), which can be easily integrated.

Long Division in Integration

When integrating rational functions, long division serves as a tool to simplify the integrand if the degree of the numerator is greater than or equal to the degree of the denominator. To use long division in integration:

• Divide the numerator by the denominator as you would in arithmetic division.
• Write the integral as the sum of the quotient plus the remainder over the original denominator.
• Integrate each part separately.

Using our example \(\int \frac{x^{2}}{x+1} dx\), long division gives us \(x - 1 + \frac{1}{x+1}\), which makes the integral straightforward to integrate.

It's crucial to perform long division correctly to avoid mistakes in the resulting integral. During integration, keeping track of this step definitely simplifies calculations.

Integrating Rational Functions

Integrating rational functions involves integrating fractions where both the numerator and the denominator are polynomials. The methods to tackle these integrals include:

• Using u-substitution when there's a function and its derivative present.
• Applying long division if the numerator's degree is at least as large as the denominator's degree.
• Factoring the denominator and using partial fractions when applicable.

For an integral like \(\int \frac{x^{2}}{x+1} dx\), both u-substitution and long division provide strategies to simplify the process. After either method, we may be left with a simpler polynomial integral and simple fractions, both of which are easier to integrate using basic integral formulas. When completing such tasks, it is important to be meticulous with algebraic manipulation to avoid errors in the final answer.