7. Antiderivatives & Indefinite Integrals

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

33. ∫ √(x² - 9)/x dx, x > 3

Evaluate the indefinite integral as a power series: $\int \frac{1}{1-t^7} dt$.

Evaluate the indefinite integral. (Remember the constant of integration.) $\int ln\left(x\right) dx$

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ √(9 - w²) dw / w²

7–64. Integration review Evaluate the following integrals.

57. ∫ dx / (x¹⸍² + x³⸍²)

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.

76. ∫ x(2x + 3)⁵ dx

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ (√2 - x) / √x dx

7–64. Integration review Evaluate the following integrals.

8. ∫ (9x - 2)^(-3) dx

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (a) ∫ e¹⁰ˣ d𝓍

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ (cos(√x))/(√x) dx

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

∫ ((1 + √x)/x)dx

Variations on the substitution method Evaluate the following integrals.                                                                                                        

 ∫ 𝓍/(√𝓍―4) d𝓍

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ x √(x² - 4) dx

6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals.

∫(𝓍 + 1)¹² d𝓍