Problem 77 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{x^{2}+2 x-1}{x^{2}+9} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \[x + \frac{7}{3} \arctan \left( \frac{x}{3} \right) - \ln |x^2 + 9| + C\].

Step 1: Simplify the Problem

Before we start integrating, we should look for ways to simplify the fraction. Notice that the numerator \(x^2 + 2x - 1\) can potentially be split into parts to ease the integration process. Let's first see if this can be expressed as a sum or difference of simpler fractions or functions.

Step 2: Express Numerator for Easier Integration

We are dealing with the fraction \(\frac{x^2 + 2x - 1}{x^2 + 9}\). We look to express the numerator in a way that may correspond to a derivative of a part of the denominator or a simpler form of integration. Notice:\[ x^2 + 2x - 1 = (x^2 + 9) - (7 - 2x) \]This allows:\[ \frac{x^2 + 2x - 1}{x^2 + 9} = 1 - \frac{7 - 2x}{x^2 + 9} \].

Step 3: Split the Integral and Solve Each Part

Express the integral now as two separate parts:\[ \int \frac{x^2 + 2x - 1}{x^2 + 9} \, dx = \int 1 \, dx - \int \frac{7 - 2x}{x^2 + 9} \, dx \]- The first integral: \[ \int 1 \, dx \] is straightforward and results in \( x + C_1 \).- The second integral: \[ \int \frac{7 - 2x}{x^2 + 9} \, dx \] can be further split using linearity: - \[ 7 \int \frac{1}{x^2 + 9} \, dx \] - \[ -2 \int \frac{x}{x^2 + 9} \, dx \].

Step 4: Evaluate \(\frac{7}{x^2+9}\) Integral

We'll evaluate \(7 \int \frac{1}{x^2 + 9} \, dx\).Recognize this as an arctangent form:\[ \int \frac{1}{x^2 + 3^2} \, dx = \frac{1}{3} \arctan \left( \frac{x}{3} \right) + C_2 \].Therefore:\[ 7 \int \frac{1}{x^2 + 9} \, dx = \frac{7}{3} \arctan \left( \frac{x}{3} \right) + C_3 \].

Step 5: Evaluate \(-\frac{2x}{x^2+9}\) Integral

We solve \(-2 \int \frac{x}{x^2 + 9} \, dx\).Use substitution: set \( u = x^2 + 9 \), then \( du = 2x \, dx \).The integral becomes:\[ -2 \int \frac{x}{x^2 + 9} \, dx = - \int \frac{du}{u} = - \ln |u| + C_4 \].Back substitute for \( u \):\[ = - \ln |x^2 + 9| + C_4 \].

Step 6: Combine Parts and Write Final Answer

Now piece together all the parts:- From the split integral: \( x \),- Plus result from \( \frac{7}{x^2+9} \): \( \frac{7}{3} \arctan \left( \frac{x}{3} \right) \),- Plus result from \(-\frac{2x}{(x^2+9)}\): \(- \ln |x^2 + 9|\).Combine these into the final solution:\[ \int \frac{x^2 + 2x - 1}{x^2 + 9} \, dx = x + \frac{7}{3} \arctan \left( \frac{x}{3} \right) - \ln |x^2 + 9| + C \],where \(C\) is the constant of integration.

Key Concepts

Integration by Substitution

Integration by substitution is like a sneak peek into transforming challenging integrals into more familiar forms. Imagine having a cake named \( \frac{x}{x^2 + 9} \) that you want to cut into simpler pieces. Here's how it goes:

• You identify a component of the integral as \( u \). Here, we choose \( u = x^2 + 9 \).
• Determine the derivative \( du \), which turns out to be \( 2x \, dx \).
• Replace parts of the integral with \( u \) and \( du \), simplifying the expression to a logarithmic form \( \ln|u| \).

This technique works like a puzzle. By fitting \( u \) and \( du \) into the expression, you simplify the integral. Once solved, remember to bring back the original variable by reversing the substitution. This method perfectly complements more complex problems where you aim to simplify the algebra involved.

Partial Fraction Decomposition

Partial fraction decomposition involves breaking down fractions into simpler, more integration-friendly pieces. Imagine you have a big jigsaw puzzle of fractions. Partial fractions help you divide it into manageable sections. In this exercise, though the main focus was simplifying the numerator \( x^2 + 2x - 1 \), the concept often applies to high-order polynomial denominators.

• Divide the original expression by the denominator if possible.
• Break down the fraction into a sum of simpler fractions where direct integration is feasible.
• This separation allows for the integration of each piece on its own, making calculations straightforward.

While our exercise didn't need full partial fraction decomposition, it still serves as a reminder of simplifying complex rationals for easier integration.

Arctangent Integration

Arctangent integration is famous for integrals involving expressions like \( \frac{1}{x^2 + a^2} \). Sounds familiar? Our exercise tackled \( \frac{7}{x^2 + 9} \). Here's how you integrate using the arctangent function:

• Recognize the integral form \( \int \frac{1}{x^2 + a^2} \, dx \).
• Recall that this equates to \( \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C \).
• Apply this result to integrate \( \frac{7}{x^2 + 9} \), obtaining \( \frac{7}{3} \arctan \left( \frac{x}{3} \right) \).

The arctangent method is neat—it transforms complex expressions into a simple arctan function by substituting the appropriate coefficient. This conversion is especially valuable in trigonometric and rational function integrals.

Logarithmic Integration

Logarithmic integration brings logarithms into play when solving integrals like \( \int \frac{1}{u} \, du \). In our exercise, we stumbled upon this while integrating \( -\frac{2x}{x^2 + 9} \). This type of integration goes like this:

• Recognize or transform the integral into the form \( \int \frac{1}{u} \, du \).
• The result is \( \ln|u| + C \), capturing the core of logarithmic integration.
• Back-substitute any substitutions made to return to the original variables.

Understanding this concept elevates your integration toolkit, especially when faced with complicated rational expressions. It highlights how logarithms can simplify seemingly convoluted problems by focusing on their derivative properties.