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

Verified step by step guidance

1. 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.
2. 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.
3. 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: \[1 = A(x^2 + 2x + 2)^2 + (Bx + C)(x + 1)(x^2 + 2x + 2) + (Dx + E)(x + 1)\]
4. 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 \(E\).
5. 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.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational functions into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of simpler rational expressions based on the factors of the denominator, including repeated and irreducible quadratic factors.

Integration of Rational Functions with Repeated Quadratic Factors

When integrating rational functions with repeated irreducible quadratic factors, the partial fractions include terms with powers of the quadratic in the denominator. Each term typically has a linear numerator, and integrating these requires techniques such as substitution and recognizing standard integral forms.

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It is especially useful after partial fraction decomposition when integrating terms involving quadratic expressions, allowing the use of standard integral formulas.