7. Antiderivatives & Indefinite Integrals

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.

4. Is a reduction formula an analytical method or a numerical method? Explain.

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

18. ∫ dx / (225 − 16x²)

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

24. ∫ dt / √(1 + 4eᵗ)