Question

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

Accepted Answer

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: $$ \frac{2y^{4}}{y^{3} - y^{2} + y - 1} = 	ext{quotient} + \frac{	ext{remainder}}{y^{3} - y^{2} + y - 1} . $$Next, factor the denominator $$ y^{3} - y^{2} + y - 1  if $$possible, to express the proper fraction $$ \frac{	ext{remainder}}{y^{3} - y^{2} + y - 1}  as a $$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.

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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Concetti chiave

Polynomial Long Division

Partial Fraction Decomposition

Integration of Rational Functions