Question 1

Evaluate the following definite integral: 
$$
\int_{3}^{7} 4x \, dx
$$

Accepted Answer

$$112$$

Question 2

Which of the following is the correct antiderivative of $$f(x) = 2x^5$ - 4x + 5$$?

Accepted Answer

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

Question 3

Evaluate the following integral: 
$$
\int x^2$ \sqrt{4 - x^3$} \, dx
$$

Accepted Answer

$$-\frac{2}{3}(4 - x^3$)^{3/2} + C$

Question 4

A company’s profit rate is given by $$P'(x) = x e$$^{-x^2}$$, where x$ is years since launch. If the profit in year 3 is $10,000, which of the following is the correct profit function?

Accepted Answer

$$P(x) = -\frac{1}{2} e$$^{-x^2}$$ + C$$

Question 5

Find the equation of the curve if the slope of the tangent is $$f'(x) = \frac{4x-20}{x^2$-10x+26}$$ and the curve passes through $$(5,6).$$

Accepted Answer

$$f(x) = 2 \ln(x^2$ - 10x + 26) + C$$, where $$C is $$chosen so $$f(5) = 6$$

Question 6

Evaluate the following definite integral: 
$$
\int_{-1}^{3} x^7$ \, dx
$$

Accepted Answer

$$\frac{3^8$ - (-1)^8$}{8}$$

Question 7

Evaluate the following integral: 
$$
\int 10x \sqrt{x^2$ + 2} \, dx
$$

Accepted Answer

$$5(x^2$$ + $$2)^{3/2} + C$

Question 8

A ball is rolled up a hill with initial velocity $$4 m/s. $$Its acceleration is $$a(t) = -3t + 6. $$How far is the ball from where it started after $$3$$ seconds?

Accepted Answer

$$13.5$$ meters from the start$

Question 9

Evaluate the following integral: 
$$
\int \frac{t^2$ + 2}{t^3$ + 6t + 3} \, dt
$$

Accepted Answer

$$\ln|t^3$ + 6t + 3| + C$$

Question 10

Find the equation of the curve if $$f'(x) = -6x^2$ + 6x - 6$$ and $$f(0) = 6.$$

Accepted Answer

$$f(x) = -2x^3$ + 3x^2$ - 6x + 6$$

Question 11

Evaluate the following integral: 
$$
\int \frac{e$$^{\sqrt{y}$$}}{2\sqrt{y}} \, dy
$$

Accepted Answer

$$e^{\sqrt{y}$$} + C$$

Question 12

A business’s profit rate is $$P'(x) = x e$$^{-x^2}$$. Which of the following best describes the long-term behavior of the profit as x$$ 	o \infty$$?

Accepted Answer

$$P(x)$$ approaches a finite value$

Question 13

Which substitution is most appropriate for evaluating $$\int x^2$ \sqrt{4 - x^3$} \, dx$$?

Accepted Answer

$$u = 4 - x^3$

Question 14

Suppose $$f'(x) = \frac{4x-20}{x^2$-10x+26}. $$What integration technique is most appropriate to find $$f(x)$$?

Accepted Answer

Substitution with $$u = x^2$ - 10x + 26$$

Question 15

A ball’s acceleration is $$a(t) = -3t + 6. If $$its initial velocity is $$4 m/s$$, what is its velocity after $$3$$ seconds?

Accepted Answer

$$-4.5 m/s$$

Question 16

Evaluate the following definite integral: 
$$
\int_{2}^{5} (x^5$ - 8) \, dx
$$

Accepted Answer

$$\frac{1}{6}(5^6$ - 2^6$) - 8(5 - 2)$$

Question 17

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

Accepted Answer

$$2\sqrt{x} + C$$

Question 18

If $$\int f(x) \, dx = F(x) + C$$, which of the following statements is always true?

Accepted Answer

$$F'(x) = f(x)$$

Business Calculus Practice: Integrals, Tangent Lines, and Applications

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