7. Antiderivatives & Indefinite Integrals

Which of the following is the $5^{th}$-degree Maclaurin polynomial for the function $f\left(x\right)=e^{-x}$?

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

Checking Antiderivative Formulas

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

∫√(2x + 1) dx = √(x² + x +C)

Suppose $F\left(x\right)$ is an antiderivative of $f\left(x\right)$. Which of the following statements about the indefinite integral $\int f\left(x\right) dx$ is true?

Applications

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

Find:

∫[f(x) + g(x)] dx

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

For the following function $f\(\left\)(x\(\right\))$, find the antiderivative $F\(\left\)(x\(\right\))$ that satisfies the given condition.

$f\(\left\)(x\(\right\))=5x^4$; $F\(\left\)(0\(\right\))=1$

Give the antiderivatives of xᵖ . For what values of p does your answer apply?

Give the antiderivatives of 1/x.

Let $h\left(x\right)$ be an antiderivative of the function $f\left(x\right)=x^3+\frac{-\sin x}{x^2}+2$. Which of the following is a correct expression for $h\left(x\right)$?

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.

csc x cot x

Suppose the graph of a function $f$ is shown below. Which of the following graphs could represent an antiderivative $F$ of $f$?

Checking Antiderivative Formulas

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

∫3(2x + 1)² dx = (2x + 1)³ + C

Checking Antiderivative Formulas

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

∫tanθ sec²θ dθ = sec³θ / 3 + C