12. Techniques of Integration

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

∫ 1 / (x⁴ + x) dx

In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.

∫ 2y⁴ / (y³ - y² + y - 1) dy

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

∫ (x² + x) / (x⁴ - 3x² - 4) dx

Evaluate the integrals in Exercises 39–54.

∫ 1 / ((x¹/³ - 1)√x) dx

(Hint: Let x = u⁶.)