7. Antiderivatives & Indefinite Integrals

If F(x) = x² - 3x + C and F (-1) = 4 , what is the value of C?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ (x²) / (4x³ + 7) dx

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

2x

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x² − 2x + 1

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x⁷

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

−3x⁻⁴

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x⁻⁴ + 2x + 3

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x⁻³/2 + x²

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

1 / x²

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

2 - 5 / x²

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

1 / 2x³

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

(3/2)√x

Applications

Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).

Find:

∫f(x) dx