Problem 73
Question
Evaluate the integrals. $$\int_{1}^{2} \frac{8 d x}{x^{2}-2 x+2}$$
Step-by-Step Solution
Verified Answer
The integral evaluates to \( 2\pi \).
1Step 1: Recognize the Form of the Integrand
The integrand \( \frac{8}{x^{2}-2x+2} \) resembles a rational function with a quadratic expression in the denominator. Notice that this quadratic expression might factor or be rewritten using a technique such as completing the square.
2Step 2: Complete the Square
Rewrite the quadratic expression in the denominator \( x^2 - 2x + 2 \) by completing the square. Start by considering \((x-1)^2 = x^2 - 2x + 1\). Thus, \( x^2 - 2x + 2 = (x-1)^2 + 1 \). This form is useful for comparison with standard integral results.
3Step 3: Use a Substitution for Integration
Let \( u = x - 1 \). Then, \( du = dx \). The limits of integration change accordingly: when \( x = 1, u = 0 \) and when \( x = 2, u = 1 \). The integral becomes \( \int_{0}^{1} \frac{8}{u^{2} + 1} \, du \).
4Step 4: Apply Standard Integration Form
The integral \( \int \frac{1}{u^2 + 1} \, du \) is a standard form with the result \( \arctan(u) \). Therefore, \( \int \frac{8}{u^{2} + 1} \, du = 8 \arctan(u) + C \).
5Step 5: Evaluate the Definite Integral
Apply the limits of integration: \(8\left[ \arctan(1) - \arctan(0) \right]\). We know that \( \arctan(1) = \frac{\pi}{4} \) and \( \arctan(0) = 0 \). Therefore, the result is \(8 \left( \frac{\pi}{4} - 0 \right) = 2\pi\).
Key Concepts
Rational Function IntegrationCompleting the SquareSubstitution MethodArctan Integration
Rational Function Integration
Integrating rational functions involves expressions that are fractions where both the numerator and the denominator are polynomials. A key aspect of these functions is to determine how to handle the polynomial in the denominator.
For example, in the expression \( \frac{8}{x^2-2x+2} \), you need to focus on the denominator. Identifying whether it can be factored or requires other techniques such as completing the square is crucial. In many cases, rational function integration calls for manipulating the denominator through algebraic techniques to simplify the integration process.
For example, in the expression \( \frac{8}{x^2-2x+2} \), you need to focus on the denominator. Identifying whether it can be factored or requires other techniques such as completing the square is crucial. In many cases, rational function integration calls for manipulating the denominator through algebraic techniques to simplify the integration process.
Completing the Square
Completing the square is a method used to reformulate quadratic expressions, making them easier to integrate. The expression \( x^2 - 2x + 2 \) is non-factorable over the real numbers, so it benefits from completing the square.
Consider how the expression can be rewritten: start with \( (x - 1)^2 = x^2 - 2x + 1 \), then add the difference \(+ 1\) to achieve \( (x - 1)^2 + 1 \). This transformation not only simplifies the expression but also aligns it with a standard form that is familiar in integration, particularly useful for expressions involving the inverse tangent function.
Consider how the expression can be rewritten: start with \( (x - 1)^2 = x^2 - 2x + 1 \), then add the difference \(+ 1\) to achieve \( (x - 1)^2 + 1 \). This transformation not only simplifies the expression but also aligns it with a standard form that is familiar in integration, particularly useful for expressions involving the inverse tangent function.
Substitution Method
The substitution method simplifies integration by changing the variable of integration. In our exercise, after completing the square, the substitution \( u = x - 1 \) transforms the integral into an easier form.
This substitution is powerful because it converts the limits of the integral: when \( x = 1 \), \( u = 0 \); when \( x = 2 \), \( u = 1 \). The transformed integral \( \int \frac{8}{u^2+1} \, du \) becomes a straightforward problem, representing a standard integral that many students will recognize as related to the arctangent function.
Substitution helps tackle integrations that seem complex at first, by making the expressions more manageable.
This substitution is powerful because it converts the limits of the integral: when \( x = 1 \), \( u = 0 \); when \( x = 2 \), \( u = 1 \). The transformed integral \( \int \frac{8}{u^2+1} \, du \) becomes a straightforward problem, representing a standard integral that many students will recognize as related to the arctangent function.
Substitution helps tackle integrations that seem complex at first, by making the expressions more manageable.
Arctan Integration
Arctan integration refers to the process of integrating functions that result in inverse trigonometric functions, specifically the arctangent. In this exercise, after the variable substitution, the integral becomes \( \int \frac{8}{u^2+1} \, du \), which is directly linked to the arctangent formula.
Recognizing standard integrals, such as \( \int \frac{1}{u^2+1} \, du = \arctan(u) + C \), is crucial for quick integration. Here, multiplying the result by 8 yields \( 8 \arctan(u) + C \).
Evaluating this integral over the limits \( u = 0 \) to \( u = 1 \) involves understanding the specific values \( \arctan(1) = \frac{\pi}{4} \) and \( \arctan(0) = 0 \), leading to the final answer \( 2\pi \). This shows how useful understanding the arctan function is when dealing with certain quadratic forms in integrals.
Recognizing standard integrals, such as \( \int \frac{1}{u^2+1} \, du = \arctan(u) + C \), is crucial for quick integration. Here, multiplying the result by 8 yields \( 8 \arctan(u) + C \).
Evaluating this integral over the limits \( u = 0 \) to \( u = 1 \) involves understanding the specific values \( \arctan(1) = \frac{\pi}{4} \) and \( \arctan(0) = 0 \), leading to the final answer \( 2\pi \). This shows how useful understanding the arctan function is when dealing with certain quadratic forms in integrals.
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