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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.5.38
Chapter 8, Problem 8.5.38

In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.
∫ 2y⁴ / (y³ - y² + y - 1) dy

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1
First, perform polynomial long division of the integrand \( \frac{2y^{4}}{y^{3} - y^{2} + y - 1} \). Since the degree of the numerator (4) is higher than the degree of the denominator (3), divide \( 2y^{4} \) by \( y^{3} \) to find the first term of the quotient.
Multiply the entire denominator \( y^{3} - y^{2} + y - 1 \) by the term found in the previous step and subtract this product from the numerator \( 2y^{4} \) to find the remainder polynomial. This remainder will have degree less than 3.
Rewrite the original integrand as the sum of the quotient polynomial plus the remainder over the original denominator: \[ \frac{2y^{4}}{y^{3} - y^{2} + y - 1} = \text{quotient} + \frac{\text{remainder}}{y^{3} - y^{2} + y - 1} \].
Next, factor the denominator \( y^{3} - y^{2} + y - 1 \) if possible, to express the proper fraction \( \frac{\text{remainder}}{y^{3} - y^{2} + y - 1} \) as a sum of partial fractions. Look for linear or quadratic factors.
Set up the partial fraction decomposition with unknown coefficients corresponding to each factor of the denominator. Then, multiply both sides by the denominator and equate coefficients of like powers of \( y \) to solve for these unknowns. Finally, integrate each term separately.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. It helps simplify the integrand by separating it into a polynomial plus a proper fraction, making the integral easier to handle. This step is essential when the degree of the numerator is equal to or greater than the denominator.
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Partial Fraction Decomposition

Partial fraction decomposition breaks down a proper rational function into a sum of simpler fractions with linear or quadratic denominators. This technique simplifies integration by allowing each term to be integrated individually. It requires factoring the denominator and expressing the fraction as a sum of unknown coefficients to solve.
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Partial Fraction Decomposition: Distinct Linear Factors

Integration of Rational Functions

Integrating rational functions often involves rewriting the integrand using long division and partial fractions. After decomposition, standard integral formulas for logarithmic and inverse trigonometric functions can be applied. Understanding these integration techniques is crucial for evaluating complex rational integrals.
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Intro to Rational Functions
Related Practice
Textbook Question

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Textbook Question

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Textbook Question

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A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral

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Textbook Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

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