7. Antiderivatives & Indefinite Integrals

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (√x(2x⁶ - 4³√)dx

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ d𝓍/[(tan⁻¹ 𝓍) (1 + 𝓍²)]

7–64. Integration review Evaluate the following integrals.

7. ∫ dx / (3 - 5x)^4

7–64. Integration review Evaluate the following integrals.

8. ∫ (9x - 2)^(-3) dx

What change of variables would you use for the integral ∫(4 - 7x)^(-6) dx?

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.