7. Antiderivatives & Indefinite Integrals

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

131. ∫ dx / (x√(1 − x⁴))

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

∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx

Evaluate the indefinite integral. (Use c for the constant of integration.)

$${\int }_{}^{}\frac{sin\left(t\right)}{1+cos\left(t\right)} dt$$

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ 𝓍³ (𝓍⁴ + 16)⁶ d𝓍

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

∫ (1/x√(36x² - 36))dx

Evaluate the integrals in Exercises 37–46.

∫(2θ + 1 + 2 cos (2θ + 1))dθ

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.

37. ∫ dx / √(x² + 10x), x >

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

∫ (⁴√x³ + √x⁵) dx

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (6 dy / √y(1 + y))

Use any method to evaluate the integrals in Exercises 55–66.

∫ x² √(1 - x²) dx

Evaluate the integrals in Exercises 1–14.

∫ √(1 - 9t²) dt

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.

11. ∫ 3u / (2u + 7) du

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

∫ (6/√(4 - 4x²))dx

Evaluate ∫ x³ √(1 - x²) dx using:

c. A trigonometric substitution.

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

∫ (3x⁵ - 5x⁹) dx