7. Antiderivatives & Indefinite Integrals

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.

92–98. Evaluate the following integrals.

94. ∫ (dt / (t³ + 1))

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (2x +1)² dx

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ ((1/x²) - (2/(x⁵⸍²))) dx

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (12/x)dx

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (x² / (x⁴ + x²)) dx

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (⁴√x³ + √x⁵) dx

1. Give some examples of analytical methods for evaluating integrals.