Question 1

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

Accepted Answer

A function $$F(x)$$ such that $$F'(x) = f(x)$$ for all $$x in $$the interval.

Question 2

What is the most general form of the indefinite integral of $$f(x)$$?

Accepted Answer

$$\int f(x) dx = F(x) + C$$, where $$C is an $$arbitrary constant.

Question 3

Evaluate $$\int 5x^4$ dx.$$

Accepted Answer

$$x^5 + C$

Question 4

Which rule allows you to integrate a sum of functions?

Accepted Answer

$$\int (af(x) + bg(x)) dx = a \int f(x) dx + b \int g(x) dx$$

Question 5

Find $$\int \frac{1}{x} dx.$$

Accepted Answer

$$\ln|x| + C$$

Question 6

Evaluate $$\int e$$^{ax}$$ dx$$ where $$a 
eq 0.$$

Accepted Answer

$$\frac{e$$^{ax}$$}{a} + C$$

Question 7

A company’s marginal revenue is $$R'(q) = 4000 - 8q$$^{-0.8}$$ (dollars per unit). What is the revenue function R(q$$)$$, assuming R(0) = 0$?

Accepted Answer

$$R(q) = 4000q + 40q$$^{0.2}$$

Question 8

An object is thrown downward from a tower with acceleration $$a(t) = -32 ft/s$^2$$ and initial velocity $$v(0) = -20 ft/s. $$What is the velocity function $$v(t)$$?

Accepted Answer

$$v(t) = -32t - 20$$

Question 9

Given $$v(t) = -32t - 20$$ and initial height $$s(0) = 1100 ft$$, what is the position function $$s(t)$$?

Accepted Answer

$$s(t) = -16t^2$ - 20t + 1100$$

Question 10

Which substitution would be most appropriate for evaluating $$\int 6x(3x^2$ + 4)^7$ dx$$?

Accepted Answer

Let $$u = 3x^2$ + 4$$

Question 11

Evaluate $$\int x^2$ \sqrt{x^3$ + 1} dx$$ using substitution.

Accepted Answer

$$\frac{2}{9}(x^3$ + 1$$)^{3/2}$$ + C$$

Question 12

Find $$\int \frac{x+3}{x^2$ + 6x} dx$$ using substitution.

Accepted Answer

$$\frac{1}{2} \ln|x^2$ + 6x| + C$$

Question 13

Evaluate $$\int \frac{2x - 3}{x^2$ - 3x} dx$$ using substitution.

Accepted Answer

$$\ln|x^2$ - 3x| + C$$

Question 14

What is the result of $$\int 3x e$$^{x^2}$$ dx$$?

Accepted Answer

$$e^{x^2} + C$

Question 15

Which substitution is appropriate for $$\int x \sqrt{1 - x} dx$$?

Accepted Answer

Let $$u = 1 - x$$

Question 16

After substituting $$u = 1 - x in$$ $$\int x \sqrt{1 - x} dx$$, what does $$x$$ become in terms of $$u$$?

Accepted Answer

$$x = 1 - u$$

Question 17

Which of the following is a correct application of the substitution rule for $$\int f(g(x))g'(x) dx$$?

Accepted Answer

Let $$u = g(x)$$, then $$du = g'(x) dx$$, so $$\int f(g(x))g'(x) dx = \int f(u) du$$

Question 18

Which of the following is NOT a typical choice for $$u in $$substitution problems?

Accepted Answer

A constant term in the integrand

Antiderivatives and Substitution: Business Calculus Study Notes

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