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Magazine Chapter 7: Problem 77
Evaluate the integrals. $$\int \frac{x^{2}+2 x-1}{x^{2}+9} d x$$
Short Answer
Expert verified
The integral evaluates to
\[x + \frac{7}{3} \arctan \left( \frac{x}{3} \right) - \ln |x^2 + 9| + C\].
Step by step solution
01
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.
02
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:\[ \frac{x^2 + 2x - 1}{x^2 + 9} = 1 - \frac{7 - 2x}{x^2 + 9} \].
03
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: - \[ 7 \int \frac{1}{x^2 + 9} \, dx \] - \[ -2 \int \frac{x}{x^2 + 9} \, dx \].
04
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:\[ 7 \int \frac{1}{x^2 + 9} \, dx = \frac{7}{3} \arctan \left( \frac{x}{3} \right) + C_3 \].
05
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 \( u \):\[ = - \ln |x^2 + 9| + C_4 \].
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( \ln|u| \).
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.
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 \( \frac{7}{3} \arctan \left( \frac{x}{3} \right) \).
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.
