7. Antiderivatives & Indefinite Integrals

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/(3³√x)

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. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

ƒ(x) = 2 sinx + 1

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫xsinx dx = -x cos x + C

Which of the following is an antiderivative of $5^x$ with respect to $x$?

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

ƒ(x) = eˣ

Which of the following is an antiderivative of $2x$?

Find a particular solution to the differential equation $\frac{dy}{dx}=2x$, given that $y\left(1\right)=5$.

Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

Finding Antiderivatives

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

(2/3)x⁻¹ᐟ³

Finding Antiderivatives

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

-(1/3)x⁻⁴ᐟ³

Which of the following is an antiderivative of $2e^x$ with respect to $x$?

Which of the following is an antiderivative of $\frac{1}{x^2}$?

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.

Which of the following is an antiderivative of $\cos x$?