7. Antiderivatives & Indefinite Integrals

Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division to show that f(x) = 4x + 1 + 1 / (x² + 1) and use this result to evaluate ∫f(x) dx.

7–64. Integration review Evaluate the following integrals.

20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx

7–64. Integration review Evaluate the following integrals.

28. ∫ (3x + 1) / √(4 - x²) dx

7–64. Integration review Evaluate the following integrals.

36. ∫ (t³ - 2) / (t + 1) dt

7–64. Integration review Evaluate the following integrals.

38. ∫ x / (x⁴ + 2x² + 1) dx

7–64. Integration review Evaluate the following integrals.

40. ∫ (1 - x) / (1 - √x) dx

7–64. Integration review Evaluate the following integrals.

47. ∫ dx / (x⁻¹ + 1)

7–64. Integration review Evaluate the following integrals.

49. ∫ √(9 + √(t + 1)) dt

7–64. Integration review Evaluate the following integrals.

57. ∫ dx / (x¹⸍² + x³⸍²)

70. Different methods Let I=∫(x+2)/(x+4)dx.

b. Evaluate I without performing long division on the integrand.

76. Different Substitutions

b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x

7–84. Evaluate the following integrals.

13. ∫ [1 / (eˣ √(1 – e²ˣ))] dx

7–84. Evaluate the following integrals.

25. ∫ [1 / (x√(1 - x²))] dx

7–64. Integration review Evaluate the following integrals.

62. ∫ (-x⁵ - x⁴ - 2x³ + 4x + 3) / (x² + x + 1) dx

68. Different methods

b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.