Problem 15 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{x^{2}-3 x+2}{x+1} d x $$

Step-by-Step Solution

Answer

\(\frac{1}{2}x^{2} - 4x + 6\ln|x + 1| + C\)

Step 1: Perform Polynomial Division

Rewrite the integrand as a quotient of two polynomials. The expression \((x^{2}-3 x+2)\) can be divided by the polynomial \(x+1\) using polynomial division. Doing so will yield: \(x - 4 + 6 / (x + 1)\).

Step 2: Separate the Terms

Separate the terms to convert complex integral into simpler forms. Now, the integral becomes the sum of simpler integrals: \(\int x dx - \int 4 dx + \int \frac{6}{x + 1} dx\).

Step 3: Perform Integration

Apply the power rule on the first integral and the rule of integrating a constant on the second integral. The formula to integrate \(\frac{1}{x + a}\) can be used directly on the third integral. This yields: \(\frac{1}{2}x^{2} - 4x + 6\ln|x + 1| + C\), where C is the constant of integration.

Key Concepts

Polynomial Division

Polynomial division is much like traditional long division. You use this method to simplify expressions into a more digestible format. This is especially useful for integration. Since the original problem provides a fraction where the numerator has a higher degree than the denominator \((x^2 - 3x + 2)/(x + 1)\), division is necessary.Here is a step-by-step approach:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this result and subtract it from the original dividend.
• Repeat these steps with the new expression until the remainder has a lower degree than the divisor.

This transforms the integrand into a combination of polynomials and simplifies the process of finding the indefinite integral.

Power Rule

The power rule is a fundamental rule in calculus used to integrate terms with powers of variables. It’s straightforward and crucial for handling polynomial terms in integrals. The power rule states:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \]where \( n eq -1 \) and \( C \) is the constant of integration.Applying this rule means you increase the exponent by one and then divide by the new exponent. It’s useful for terms like \(x-4\) in our example. It helps break down complex integrals into simpler, manageable parts.

Integrating Constants

Integrating constants is a simple but fundamental process in calculus. When you have an integral involving a constant, like \-4\, the process is straightforward:\[ \int a \, dx = ax + C, \]where \( a \) is a constant and \( C \) is the constant of integration.For example, with the term \(-4\), integrating it yields \-4x\. This highlights how constants interact differently than variable terms, providing a simple sum once integrated.

Natural Logarithm Integration

The natural logarithm integration rule helps when integrating terms like \(\frac{1}{x + a}\). The integration process follows this rule:\[ \int \frac{1}{x + a} \, dx = \ln|x + a| + C. \]This method is crucial when dealing with integrals resulting from polynomial division, where constants are added or subtracted from variables. It uses the properties of logarithms to manage variable-based denominators, turning fractional expressions into a more straightforward logarithmic form. In our example, this aids in effectively integrating \(\frac{6}{x+1}\).