Problem 13 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{x^{2}+2 x+3}{x^{3}+3 x^{2}+9 x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\frac{x^{2}+2 x+3}{x^{3}+3 x^{2}+9 x}\) dx is \(\frac{1}{2} x^{2} + 2x + \frac{3}{2} ln |x^{2}+3 x+9| + C\).

Step 1: Simplify the integrand

Divide each term in the numerator by \(x\), so your integral is simplified to \(\int \frac{x+2+\frac{3}{x}}{x^{2}+3 x+9} dx \).

Step 2: Break down the integral

Now, break the integral down into simpler term-by-term integrals. The integral can be rewritten as \(\int x dx + \int 2 dx + \int \frac{3}{x^{2}+3 x+9} dx \).

Step 3: Solve the integrals

Now solve each simpler integral independently. \(\int x dx \) yields \(\frac{1}{2} x^{2}\). \(\int 2 dx \) yields \(2x\). And \(\int \frac{3}{x^{2}+3 x+9} dx \) yields \(\frac{3}{2} ln |x^{2}+3 x+9|\).

Step 4: Combine solutions

Combine the results of the integrals to form your final solution: \(\frac{1}{2} x^{2} + 2x + \frac{3}{2} ln |x^{2}+3 x+9| + C\), where C represents the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are a set of methods used to find the integral of a function, which is the reverse of differentiation. In this exercise, different methods are applied to solve the indefinite integral. Let's break down these techniques:

• Polynomial Division Techniques: Integral expressions often need simplification. This includes dividing polynomials, as seen when the numerator and denominator terms are simplified by dividing each by \(x\).
• Decomposition: Breaking down the integral into manageable parts. The original expression \( \int \frac{x^2 + 2x + 3}{x^3 + 3x^2 + 9x} dx \) is split into parts that allow the application of different techniques.

This use of various methods makes complex integrals more tractable, allowing each smaller part to be integrated using simpler techniques.

Polynomial Division

Polynomial division is a critical step in simplifying an integrand before applying integration. It involves dividing one polynomial by another. Here, this was used to ease the complexity of the original integrand. By dividing the terms of the numerator by \(x\), we achieved:

• Division: \(\frac{x^2 + 2x + 3}{x^3 + 3x^2 + 9x}\) was simplified by dividing the numerator terms until a simpler form \(\frac{x+2+rac{3}{x}}{x^{2}+3x+9}\).
• Result: This allows the integral to be broken down into parts that are more straightforward to handle using standard integration techniques.

The simplification resulting from polynomial division often makes the subsequent integration steps more straightforward.

Logarithmic Integration

Logarithmic integration is a technique typically used when integrating functions that yield a logarithmic result. In this exercise, we encounter it when integrating the term \(\frac{3}{x^{2}+3x+9}\):

• This requires recognizing a pattern or simplifying the expression into a form that fits the standard logarithmic integration formula \(\int \frac{1}{u} du = \ln |u| + C\).
• Here, the integral \(\int \frac{3}{x^2 + 3x + 9} dx\) translates through this method into \(\frac{3}{2} \ln |x^2 + 3x + 9|\) by considering a substitution or pattern recognition.

Logarithmic integration is powerful for integrals that hint at a natural logarithmic outcome due to their form.

Constant of Integration

The constant of integration is a vital concept whenever indefinite integrals are calculated. When integrating a function to find its antiderivative, we include a term \(+ C\) called the "constant of integration." Here's why it's essential:

• Basic Concept: Indefinite integration doesn't determine a single antiderivative but a family of functions. Without the constant \(C\), each indefinite integral would imply a specific solution which could be misleading.
• Inclusion: It is included in the final result, as shown: \( \frac{1}{2} x^2 + 2x + \frac{3}{2} \ln |x^2 + 3x + 9| + C\).
• Importance in calculus: This constant bridges different particular solutions of the function that share the same rate of change.

Recognizing the necessity of \(C\) is crucial for complete understanding and accurate results in indefinite integrations.