12. Techniques of Integration

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (2x + 1) / (x² - 7x + 12) dx

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (y + 4) / (y² + y) dy from 1/2 to 1

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x + 3) / (2x³ - 8x) dx

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

In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x² dx) / ((x - 1)(x² + 2x + 1))

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (8x² + 8x + 2) / (4x² + 1)² dx

Expand the quotients in Exercises 1–8 by partial fractions.

(t⁴ + 9) / (t⁴ + 9t²)

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [t / (t⁴ − t² − 2)] dt

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [(x³ + 1) / (x³ − x)] dx

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [1 / (x (1 + ∛x))] dx

Evaluate the integrals in Exercises 39–54.

∫ 1 / (cos θ + sin 2θ) dθ

Use any method to evaluate the integrals in Exercises 55–66.

∫ (x + 2) / (x³ - 2x² - 3x) dx

Use any method to evaluate the integrals in Exercises 55–66.

∫ 2^x / (2²x + 2^x - 2) dx

7. What is the goal of the method of partial fractions?