Textbook Question
7–84. Evaluate the following integrals.
22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.5.47
23-64. Integration Evaluate the following integrals.
47. ∫ (x³ - 10x² + 27x)/(x² - 10x + 25) dx
Verified step by step guidance1
First, observe the integral \( \int \frac{x^{3} - 10x^{2} + 27x}{x^{2} - 10x + 25} \, dx \). Notice that the denominator \( x^{2} - 10x + 25 \) can be factored or recognized as a perfect square. Specifically, \( x^{2} - 10x + 25 = (x - 5)^{2} \).
Next, perform polynomial division because the degree of the numerator (3) is higher than the degree of the denominator (2). Divide \( x^{3} - 10x^{2} + 27x \) by \( (x - 5)^{2} \) to rewrite the integrand as a polynomial plus a proper rational function.
After the division, express the integrand as \( Q(x) + \frac{R(x)}{(x - 5)^{2}} \), where \( Q(x) \) is the quotient polynomial and \( R(x) \) is the remainder polynomial with degree less than 2.
Then, split the integral into two parts: \( \int Q(x) \, dx + \int \frac{R(x)}{(x - 5)^{2}} \, dx \). The first integral is straightforward since \( Q(x) \) is a polynomial.
For the second integral, use substitution \( u = x - 5 \) to simplify the denominator and then integrate the resulting rational function by breaking it into simpler terms if necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Long Division
When the degree of the numerator is equal to or greater than the denominator in a rational function, polynomial long division is used to simplify the integrand. This process rewrites the integrand as a polynomial plus a proper fraction, making the integral easier to evaluate.
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Taylor Polynomials
Integration of Rational Functions
Integrating rational functions often involves breaking them into simpler parts, such as polynomials and proper fractions. After simplification, standard integration techniques like substitution or partial fractions can be applied to evaluate the integral.
Recommended video:
6:04Intro to Rational Functions
Recognizing Perfect Square Trinomials
Identifying perfect square trinomials in the denominator, such as (x - 5)², helps simplify the integral. This recognition allows substitution methods or rewriting the integrand in a form that is easier to integrate.
Recommended video:
05:22Completing the Square to Rewrite the Integrand
Related Practice
Textbook Question
9–61. Trigonometric integrals Evaluate the following integrals.
45. ∫ sec²x tan¹ᐟ²x dx
Textbook Question
23-64. Integration Evaluate the following integrals.
50. ∫ 8(x² + 4)/[x(x² + 8)] dx
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Textbook Question
66. Integrating derivatives
Use integration by parts to show that if f' is continuous on [a, b], then
∫[a to b] f(x)f'(x) dx = (1/2)[f(b)² - f(a)²]
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Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
54. ∫ y⁴/(1 + y²) dy
Textbook Question
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt