Problem 85 - Calculus and its applications

Question

Solve each integral. Each can be found using rules developed in this section, but some algebra may be required. $$ \int \frac{x^{2}-1}{x+1} d x $$

Step-by-Step Solution

Answer

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

Step 1: Perform Polynomial Long Division

Since the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator, apply polynomial long division. Divide \(x^2 - 1\) by \(x + 1\). The result of this division is \(x - 1\) with a remainder of \(0\).

Step 2: Express the Integral using the Result of Long Division

Rewrite the integral \(\int \frac{x^2 - 1}{x + 1} \, dx\) as \(\int (x - 1) \, dx\) based on the result of the polynomial division.

Step 3: Integrate the Simplified Expression

Apply the power rule of integration to find the antiderivative. The integral of \(x\) is \(\frac{x^2}{2}\) and the integral of \(-1\) is \(-x\). Thus, \(\int (x - 1) \, dx = \frac{x^2}{2} - x + C\), where \(C\) is the constant of integration.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to the long division you learned with numbers. In our exercise, we have the polynomial \(x^2 - 1\) and we need to divide it by \(x + 1\). Here's how the process works:

• First, divide the first term of the numerator \(x^2\) by the first term of the denominator \(x\), which gives \(x\).
• Multiply the entire divisor \(x + 1\) by this result \(x\) and subtract this product from the original numerator \(x^2 - 1\).
• This subtraction gives you a new expression: \(-x - 1\).
• Repeat the process with this new expression: divide \(-x\) by \(x\) to obtain \(-1\), then multiply \(x + 1\) by \(-1\) and continue the division.
• The final result gives us \(x - 1\) with a remainder of zero, which means the original fraction simplifies perfectly to a linear polynomial.

Understanding polynomial long division is crucial because it allows you to simplify polynomials before integrating or solving any related algebraic equations.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function. When dealing with the integral \(\int (x - 1) \, dx\), we are essentially finding a function whose derivative equals \(x - 1\).
To find an antiderivative, you apply integration rules such as the power rule, which helps to reverse the process of differentiation.
Finding an antiderivative is deeply tied to understanding how integration and differentiation interact, making it an essential concept in calculus. It results in the expression \(\frac{x^2}{2} - x + C\), where \(C\) is a constant that represents any constant function.

Power Rule of Integration

The power rule of integration is an essential technique used to solve integrals involving polynomial expressions. This rule states:

• For a function \(x^n\), its integral is \(\frac{x^{n+1}}{n+1} + C\), provided that \(n eq -1\).
• This allows you to easily find the antiderivative of common polynomial terms by increasing the exponent by one and dividing by the new exponent.

Applying the power rule to our simplified integral \(\int (x - 1) \, dx\), we treat \(x\) and the constant \(-1\) separately.

• The antiderivative of \(x\) is found by applying the rule: \(\frac{x^2}{2}\).
• For the constant \(-1\), its antiderivative is \(-x\) because the integral of a constant \(a\) is \(ax + C\).

Mastering the power rule of integration is vital for solving basic and complex integrals, as it often serves as the foundation upon which more advanced techniques are built.