Q: Which rule allows you to find the antiderivative of a sum of functions?

Sum or Difference Rule: $$\\int [f(x) + g(x)] dx = \\int f(x) dx + \\int g(x) dx$$

Q: Find the indefinite integral: $$\\int (2x^3$ - 5x + 6) dx$$

$$\\frac{2}{4}x^4$ - \\frac{5}{2}x^2$ + 6x + C$$

Q: If $$f(x) = 5x^2$ - 3x + 2$$, what is $$\\int f(x) dx$$?

$$\\frac{5}{3}x^3$ - \\frac{3}{2}x^2$ + 2x + C$$

Q: A business finds its marginal profit function is $$P'(x) = \\sqrt{x}$$, where $$x is $$the number of units sold. If the profit is $$1000$$ when $$x = 0$$, what is the profit function $$P(x)$$?

$$\\frac{2}{3}x$$^{3/2}$$ + 1000$$

Q: If $$f(x) = 7x^2$ - 4x + 9$$, what is $$\\int f(x) dx$$?

$$\\frac{7}{3}x^3$ - 2x^2$ + 9x + C$$

Question 1

Which of the following best describes an antiderivative of a function $$f(x)$$?

Accepted Answer

A function $$F(x)$$ such that $$F'(x) = f(x).$$

Question 2

What is the general form of the indefinite integral of $$f(x) = 3x^2$?

Accepted Answer

$$rac{3}{3}x$$^{3}$$ + C$$

Question 3

Which rule allows you to find the antiderivative of a sum of functions?

Accepted Answer

Sum or Difference Rule: $$\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$$

Question 4

Find the indefinite integral: $$\int (2x^3$ - 5x + 6) dx$$

Accepted Answer

$$\frac{2}{4}x^4$ - \frac{5}{2}x^2$ + 6x + C$$

Question 5

Which of the following is the correct antiderivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$\frac{1}{2}e$$^{2x}$$ + C$$

Question 6

A company’s marginal cost function is $$C'(x) = 2x + 5. If $$the fixed cost is $$10$$, what is the total cost function $$C(x)$$?

Accepted Answer

$$x^2 + 5x + 10$

Question 7

Find $$F(x) if$$ $$F'(x) = 4x^3$ - 2x + 7$$ and $$F(0) = 5.$$

Accepted Answer

$$x^4$$ - $$x^2 + 7x + 5$

Question 8

Which of the following is the correct indefinite integral for $$\int \frac{1}{x} dx$$?

Accepted Answer

$$\ln|x| + C$$

Question 9

If $$f(x) = 5x^2$ - 3x + 2$$, what is $$\int f(x) dx$$?

Accepted Answer

$$\frac{5}{3}x^3$ - \frac{3}{2}x^2$ + 2x + C$$

Question 10

A business finds its marginal profit function is $$P'(x) = \sqrt{x}$$, where $$x is $$the number of units sold. If the profit is $$1000$$ when $$x = 0$$, what is the profit function $$P(x)$$?

Accepted Answer

$$\frac{2}{3}x$$^{3/2}$$ + 1000$$

Question 11

Which of the following is the correct antiderivative for $$f(x) = a^x$$, where $$a \u003e 0$$ and $$a \neq 1$$?

Accepted Answer

$$\frac{a^x}{\ln a} + C$$

Question 12

Find the indefinite integral: $$\int (x$$^{-2}$$ + 4x^3$) dx$$

Accepted Answer

$$-x^{-1}$$ + $$x^4 + C$

Question 13

Suppose $$C'(x) = 3x^2$ and C(2) = 10$. What is C(x$$)$$?

Accepted Answer

Question 14

Which of the following is the correct indefinite integral for $$\int \frac{1}{x} dx$$?

Accepted Answer

$$\ln|x| + C$$

Question 15

If $$f(x) = 2e$$^{3x}$$, what is $$\int f(x) dx$$?

Accepted Answer

$$\frac{2}{3}e$$^{3x}$$ + C$$

Question 16

A company’s marginal revenue function is $$R'(x) = 4x - 7. If$$ $$R(0) = 20$$, what is the revenue function $$R(x)$$?

Accepted Answer

$$2x^2 - 7x + 20$

Question 17

Find the indefinite integral: $$\int (5x^4$ - 2x + 1) dx$$

Accepted Answer

$$x^5$$ - $$x^2 + x + C$

Question 18

Which of the following is the correct antiderivative for $$f(x) = \frac{1}{x}$$?

Accepted Answer

$$\ln|x| + C$$

Question 19

If $$f(x) = 7x^2$ - 4x + 9$$, what is $$\int f(x) dx$$?

Accepted Answer

$$\frac{7}{3}x^3$ - 2x^2$ + 9x + C$$

Section 7.1: Antiderivatives – Business Calculus Study Notes

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