7. Antiderivatives & Indefinite Integrals

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.

∫ [y / √(16 − y²)] dy

Evaluate the indefinite integral: ${4x}^2+5x+1 dx$

Evaluate the indefinite integral. (Use $c$ for the constant of integration.) $\int x(x^2+x+2)dx$

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

∫ (4√x - (4 /√x)) dx

7–84. Evaluate the following integrals.

64. ∫ (ln(ax))/x dx, where a ≠ 0

Evaluate the integral. (Use c for the constant of integration.) $\int 2{\sin }^2\left(x\right){\cos }^3\left(x\right)dx$

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

 ∫ 2𝓍(𝓍² ― 1)⁹⁹ d𝓍

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

40. ∫ (x² - 4)/(x + 4) dx

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

 ∫ [ 1/(10𝓍―3) d𝓍

Evaluate the integrals in Exercises 97–110.

97. ∫ 3x^(√3) dx

Evaluate the indefinite integral as a power series: $\int \frac{1}{1-t^9} dt$.

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

∫ (sec² x - 1) dx

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(g) ∫ ƒ' (g(𝓍))g' (𝓍) d(𝓍) = ƒ(g(𝓍)) + C .

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

∫ (x² / (x⁴ + x²)) dx

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

∫ (csc² Θ + 2Θ² - 3Θ) dΘ