Question

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

Accepted Answer

First, recognize that the denominator can be factored further. Notice that $$x^{2} + 2x + 1$ is a perfect square trinomial, so rewrite it as (x$$ + $$1)^{2}. $$Thus, the integral becomes $$\int \frac{x$$^{2}$$}{(x - 1)(x + 1$$)^{2}$$} \, dx. $$Set up the partial fraction decomposition for the integrand. Since the denominator has a linear factor $$(x - 1)$$ and a repeated linear factor $$(x + 1$$)^{2}$$, express the fraction as: $$\frac{$$x^{2}$$}{(x - 1)(x + $$1)^{2}$$} = \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{C}{(x + $$1)^{2}$$}$$. Multiply both sides of the equation by the denominator (x - 1)(x$$ + $$1)^{2} to $$clear the fractions, resulting in: $$x^{2} = A(x$$ + $$1)^{2} + B(x - 1)(x + 1) + C(x - 1$$)$$. Expand the right-hand side and collect like terms in powers of x$. This will give you a polynomial equation in x$ where the coefficients of corresponding powers of x$ on both sides must be equal. Use this to set up a system of equations to solve for A$, B$, and C$. Once you find the values of A$, B$, and C$, rewrite the integral as the sum of simpler integrals: $$\int \frac{A}{x - 1} \, dx + \int \frac{B}{x + 1} \, dx + \int \frac{C}{(x + $$1)^{2}$$} \, dx$$. Then, integrate each term separately using standard integral formulas.

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

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Concetti chiave

Partial Fraction Decomposition

Integration of Rational Functions

Handling Repeated and Quadratic Factors