In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x³ dx) / (x² - 2x + 1) from -1 to 0

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Recognize that the denominator \(x^{2} - 2x + 1\) can be factored as \((x - 1)^{2}\) since it is a perfect square trinomial.

Rewrite the integrand as \(\frac{x^{3}}{(x - 1)^{2}}\) and consider expressing it as a sum of partial fractions. Since the denominator is a repeated linear factor, the partial fraction decomposition will have the form \(\frac{A}{x - 1} + \frac{B}{(x - 1)^{2}}\).

Set up the equation: \(\frac{x^{3}}{(x - 1)^{2}} = \frac{A}{x - 1} + \frac{B}{(x - 1)^{2}}\). Multiply both sides by \((x - 1)^{2}\) to clear the denominators, resulting in \(x^{3} = A(x - 1) + B\).

Expand and simplify the right side: \(A(x - 1) + B = Ax - A + B\). Equate coefficients of powers of \(x\) on both sides to solve for constants \(A\) and \(B\).

Once \(A\) and \(B\) are found, rewrite the integrand as the sum of partial fractions and integrate term-by-term over the interval from \(-1\) to \(0\). Remember to apply the definite integral limits after integrating.

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with simpler denominators, often linear or quadratic factors. This method is especially useful when the denominator can be factored into repeated or distinct linear terms.

Polynomial Division

Polynomial division is used when the degree of the numerator is equal to or greater than the degree of the denominator. It simplifies the integrand by dividing the polynomials to rewrite the integrand as a polynomial plus a proper fraction. This step is essential before applying partial fraction decomposition if the integrand is an improper rational function.

Definite Integration

Definite integration calculates the exact area under a curve between two limits. After simplifying the integrand, the integral is evaluated by finding the antiderivative and then applying the Fundamental Theorem of Calculus to compute the difference between the values at the upper and lower bounds. This process yields a numerical value representing the integral over the specified interval.

In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.∫ (x³ dx) / (x² - 2x + 1) from -1 to 0