Problem 81
Question
Evaluate the integrals in Exercises \(71-84\) $$ \int \frac{x^{2}+2 x-1}{x^{2}+9} d x $$
Step-by-Step Solution
Verified Answer
\( x + \ln(x^2 + 9) - \frac{7}{3}\arctan\left(\frac{x}{3}\right) + C. \)
1Step 1: Identify the Integral Type
The integral given is \( \int \frac{x^{2}+2x-1}{x^{2}+9} \, dx \). This is a rational function integral since both the numerator and the denominator are polynomials.
2Step 2: Simplify the Expression
Divide the numerator by the denominator to simplify the integral. Here, the numerator is of the same degree as the denominator. Use polynomial long division to rewrite the integral if necessary.
3Step 3: Perform Polynomial Long Division
Divide \( x^2 + 2x - 1 \) by \( x^2 + 9 \). The result is 1 with a remainder of \(-7 + 2x \). Therefore, the integral becomes: \( \int 1 \, dx + \int \frac{2x - 7}{x^2 + 9} \, dx \).
4Step 4: Integrate the Simplified Terms
Integrate the result of the division:- \( \int 1 \, dx \) is simply \( x \).- \( \int \frac{2x}{x^2 + 9} \, dx \) can be solved using substitution.For \( \int \frac{2x}{x^2 + 9} \, dx \), let \( u = x^2 + 9 \), so \( du = 2x \, dx \). The integral becomes \( \int \frac{1}{u} \, du = \ln|u| + C = \ln(x^2 + 9) + C \).
5Step 5: Integrate the Constant Term in the Remainder
Consider the remaining term from the division: \( \int \frac{-7}{x^2 + 9} \, dx \), which resembles \( \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) \).Here, \( a = 3 \), so the integral is: \( -7 \cdot \frac{1}{3} \arctan\left(\frac{x}{3}\right) = -\frac{7}{3}\arctan\left(\frac{x}{3}\right) \).
6Step 6: Combine the Results of the Integration Steps
Combine all parts from the previous steps:\[ x + \ln(x^2 + 9) - \frac{7}{3}\arctan\left(\frac{x}{3}\right) + C \]This is the final expression of the integral \( \int \frac{x^{2}+2x-1}{x^{2}+9} \, dx \).
Key Concepts
Polynomial Long DivisionSubstitution MethodTrigonometric IntegrationIntegration Techniques
Polynomial Long Division
Polynomial long division is a handy technique used to simplify the division of one polynomial by another. It works similarly to numerical long division. In the context of integration, particularly for rational functions where the degree of the numerator is equal to or greater than that of the denominator, it helps break down the integral into simpler, more manageable parts.
For the integral \( \int \frac{x^{2}+2x-1}{x^{2}+9} \, dx \), the polynomial \(x^2 + 2x - 1\) is divided by \(x^2 + 9\). The result of this division is \(1\) with a remainder of \(2x - 7\). Thus, the original integral is simplified to \( \int 1 \, dx + \int \frac{2x - 7}{x^2 + 9} \, dx \).
This step effectively reduces the complexity, making subsequent integration steps more straightforward. It turns a complicated rational function into a simpler one where basic integration techniques can be applied.
For the integral \( \int \frac{x^{2}+2x-1}{x^{2}+9} \, dx \), the polynomial \(x^2 + 2x - 1\) is divided by \(x^2 + 9\). The result of this division is \(1\) with a remainder of \(2x - 7\). Thus, the original integral is simplified to \( \int 1 \, dx + \int \frac{2x - 7}{x^2 + 9} \, dx \).
This step effectively reduces the complexity, making subsequent integration steps more straightforward. It turns a complicated rational function into a simpler one where basic integration techniques can be applied.
Substitution Method
The substitution method is an essential integration technique used to simplify the integration process by changing variables. It is particularly useful when dealing with integrals that contain composite functions.
Consider the integral \(\int \frac{2x}{x^2 + 9} \, dx\). By letting \( u = x^2 + 9 \), we recognize that \( du = 2x \, dx \). This substitution transforms the integral into \( \int \frac{1}{u} \, du \). This is a renowned standard form, leading directly to the natural logarithm function: \( \ln|u| + C = \ln(x^2 + 9) + C \).
Through substitution, the integration of a complicated expression with \(x\) becomes the integration of a much simpler function in terms of \(u\). This technique is invaluable in evaluating integrals that present as fractions or compositions of functions, facilitating easier computation.
Consider the integral \(\int \frac{2x}{x^2 + 9} \, dx\). By letting \( u = x^2 + 9 \), we recognize that \( du = 2x \, dx \). This substitution transforms the integral into \( \int \frac{1}{u} \, du \). This is a renowned standard form, leading directly to the natural logarithm function: \( \ln|u| + C = \ln(x^2 + 9) + C \).
Through substitution, the integration of a complicated expression with \(x\) becomes the integration of a much simpler function in terms of \(u\). This technique is invaluable in evaluating integrals that present as fractions or compositions of functions, facilitating easier computation.
Trigonometric Integration
Trigonometric integration is a method for integrating functions that involve trigonometric functions. It often involves recognizing and applying trigonometric identities or substitutions to simplify the integral.
In the original problem, we handle the term \( \int \frac{-7}{x^2 + 9} \, dx \) by relating it to a trigonometric standard integral form. Specifically, it resembles \( \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C \).
Here, \( a = 3 \), thus the integral becomes: \( -7 \cdot \frac{1}{3} \arctan\left(\frac{x}{3}\right) = -\frac{7}{3}\arctan\left(\frac{x}{3}\right) \).
Trigonometric integration allows us to deal with integrals involving square of variables plus constants, which are prevalent in problems linking algebraic expressions with trigonometric functions.
In the original problem, we handle the term \( \int \frac{-7}{x^2 + 9} \, dx \) by relating it to a trigonometric standard integral form. Specifically, it resembles \( \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C \).
Here, \( a = 3 \), thus the integral becomes: \( -7 \cdot \frac{1}{3} \arctan\left(\frac{x}{3}\right) = -\frac{7}{3}\arctan\left(\frac{x}{3}\right) \).
Trigonometric integration allows us to deal with integrals involving square of variables plus constants, which are prevalent in problems linking algebraic expressions with trigonometric functions.
Integration Techniques
Integration techniques are crucial strategies used to solve integrals, especially when dealing with complex functions. Common techniques include substitution, partial fraction decomposition, integration by parts, and the ones applied in this problem - polynomial long division and trigonometric integration.
The objective of these techniques is to transform a difficult integral into a simpler form that is easier to manage. In our exercise, using polynomial long division simplified the rational function into easier components, while the substitution method turned a composite function into a straightforward logarithmic integral.
Moreover, trigonometric integration handled expressions involving terms like \(x^2 + a^2\), using trigonometric identities to simplify the process. Employing these techniques makes the integration process more efficient and feasible, enabling the solving of a broader range of integrals that might initially seem challenging.
The objective of these techniques is to transform a difficult integral into a simpler form that is easier to manage. In our exercise, using polynomial long division simplified the rational function into easier components, while the substitution method turned a composite function into a straightforward logarithmic integral.
Moreover, trigonometric integration handled expressions involving terms like \(x^2 + a^2\), using trigonometric identities to simplify the process. Employing these techniques makes the integration process more efficient and feasible, enabling the solving of a broader range of integrals that might initially seem challenging.
Other exercises in this chapter
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