Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (1 / (cos² x tan x)) dx from π/3 to π/4
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.5.28
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ 1 / (x⁴ + x) dx
Verified step by step guidance1
Start by factoring the denominator of the integrand. The denominator is \(x^{4} + x\), which can be factored by taking out the common factor \(x\):
\[x^{4} + x = x(x^{3} + 1)\]
Next, factor the cubic term \(x^{3} + 1\) using the sum of cubes formula:
\[a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\]
Here, \(a = x\) and \(b = 1\), so
\[x^{3} + 1 = (x + 1)(x^{2} - x + 1)\]
Rewrite the integrand using the full factorization:
\[\frac{1}{x^{4} + x} = \frac{1}{x(x + 1)(x^{2} - x + 1)}\]
Set up the partial fraction decomposition for the integrand:
\[\frac{1}{x(x + 1)(x^{2} - x + 1)} = \frac{A}{x} + \frac{B}{x + 1} + \frac{Cx + D}{x^{2} - x + 1}\]
where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.
Multiply both sides by the denominator \(x(x + 1)(x^{2} - x + 1)\) to clear the fractions, then equate coefficients of corresponding powers of \(x\) to form a system of equations. Solve this system to find the values of \(A\), \(B\), \(C\), and \(D\). Once found, integrate each term separately to evaluate the integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to express a complex rational function as a sum of simpler fractions. This method is especially useful for integrating rational functions by breaking them into terms that are easier to integrate individually.
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Partial Fraction Decomposition: Distinct Linear Factors
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of its factors. For partial fractions, factoring the denominator into linear and irreducible quadratic factors is essential to set up the correct form of the decomposition.
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07:00Taylor Polynomials
Integration of Rational Functions
Integrating rational functions often requires rewriting the integrand into simpler parts via partial fractions. Each simpler fraction corresponds to a standard integral form, such as logarithmic or arctangent functions, facilitating the evaluation of the integral.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x² √(4x - 9))
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ cos^(-1)(√x) / √x dx
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (1 - x²)^(1/2) / x⁴ dx
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(7 - x²))