12. Techniques of Integration

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

12. ∫ (8x + 5)/(2x² + 3x + 1) dx

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

63. ∫ dx/(x² - 2x - 15)

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

65. ∫ (from 0 to 1) dy/((y + 1)(y² + 1))

96. Challenge

Show that with the change of variables u = √tan x, the integral

∫ √tan x dx

can be converted to an integral amenable to partial fractions. Evaluate

∫[0 to π/4] √tan x dx.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ e^t dt / (e^(2t) + 3e^t + 2)

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

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.

∫ [(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

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (x + 1) / (x⁴ − x³) dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ 9 dv / (81 − v⁴)

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (z + 1) / [z²(z² + 4)] dz

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

∫ [1 / √(e^s + 1)] ds