12. Techniques of Integration

Express the rational function as a sum or difference of two simpler fractions. Use a system of equations.

$\frac{1}{(2x+1)(2x-1)}$

Express the rational function as a sum or difference of two simpler fractions. Use a system of equations.

$\frac{4x^2-21x-40}{x^3-x^2-20x}$

Give the partial fraction decomposition for the following expression using strategic substitutions for $x$.

$\frac{4x^2-8x-8}{x(x-2)(x+2)}$

Express the rational function as a sum or difference or simpler rational expressions.

$\frac{4}{x^3-3x^2-x+3}$

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

21. ∫ cos x / (sin² x + 2 sin x) dx

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

34. ∫ dx / (x(x¹⁰ + 1))

7–84. Evaluate the following integrals.

45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.

∫ (5x² + 18x + 20) / [(2x + 3)(x² + 4x + 8)] dx