7. Antiderivatives & Indefinite Integrals

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

∫ ((1/x²) - (2/(x⁵⸍²))) dx

Find the following indefinite integral.

$\int 100x^{99}dx$

Evaluate the integrals in Exercises 1–14.

∫ 5 dx / √(25x² - 9), where x > 3/5

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(3t² + t/2) dt

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

∫(e³ˣ ⁺¹ d𝓍

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

48. ∫ √(9 - 4x²) dx

76. Different Substitutions

b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x

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.

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

∫ dx / (x² √(x² + 1))

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

∫ (1 - r²)^(5/2) / r⁸ dr

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

∫ (eˣ⁺²) dx

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(cscθ cotθ) / 2 dθ

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.

∫ (dθ / √(2θ - θ²))

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

 ∫ sin 𝓍 sec⁸ 𝓍 d𝓍

Evaluate the indefinite integral: ${\int }_{}^{}7x^2+49dx$.