Question

23-64. Integration Evaluate the following integrals.62. ∫ 1/[(x + 1)(x² + 2x + 2)²] dx

Accepted Answer

Start by recognizing that the integrand is a rational function with a linear factor $$(x + 1)$$ and a repeated quadratic factor $$(x^2 + 2x$$ + $$2)^2 in $$the denominator. This suggests using partial fraction decomposition to break the integrand into simpler fractions. Set up the partial fraction decomposition in the form: $$\frac{1}{(x + 1)(x^2 + 2x + 2)^2} = \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 2x + 2} + \frac{Dx + E}{(x^2 + 2x + 2)^2}$$ where $$A$$, $$B$$, $$C$$, $$D$$, and $$E$$ are constants to be determined. Multiply both sides of the equation by the common denominator $$(x + 1)(x^2$ + 2x + 2)^2$ to clear the denominators, resulting in a polynomial identity: 1 = A(x^2 + 2x + 2)^2 + (Bx + C)(x + 1)(x^2 + 2x + 2) + (Dx + E)(x + 1$$)$$ Expand the right-hand side polynomial and collect like terms by powers of x$. Then, equate the coefficients of corresponding powers of x$ on both sides to form a system of linear equations for A$, B$, C$, D$, and E$. Solve the system of equations to find the values of A$, B$, C$, D$, and E$. Once these constants are found, rewrite the integral as a sum of simpler integrals involving these partial fractions, which can then be integrated using standard techniques such as substitution and recognizing derivatives of inverse trigonometric functions.

23-64. Integration Evaluate the following integrals.
62. ∫ 1/[(x + 1)(x² + 2x + 2)²] dx

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Key Concepts

Partial Fraction Decomposition

Integration of Rational Functions with Repeated Quadratic Factors

Substitution Method in Integration