Question 1

Evaluate the integral $$\int (3x$$^{1/3}$$ - 4x$$^{-1/3}$$ + 6) \, dx.$$

Accepted Answer

$$\frac{9}{4}x$$^{4/3}$$ - 6x$$^{2/3}$$ + 6x + C$$

Question 2

A medication is administered such that its rate of absorption in the bloodstream is modeled by $$a(t) = 2e^t - 12$$, where $$t is in $$hours. If the initial velocity of absorption is $$v(0) = 1$$ and the initial amount absorbed is $$s(0) = 3$$, what is the function for the total amount absorbed $$s(t)$$?

Accepted Answer

$$s(t) = 2e^t - 6t^2$ - t + 1$$

Question 3

Evaluate the definite integral $$\int_{-1}^{2} (2x - 12x^3$) \, dx.$$

Accepted Answer

$$7/2$$

Question 4

A car's acceleration is constant at $$a(t) = -32 ft/s$^2. If $$its velocity at $$t=1 is$$ $$20 ft/s $$and its position at $$t=2 is$$ $$0$$, what is its position function $$s(t)$$?

Accepted Answer

$$s(t) = -16t^2$ + 52t - 40$$

Question 5

Which of the following is the correct antiderivative of $$\int 4x \sqrt{x^2$ + 8} \, dx$$?

Accepted Answer

$$\frac{4}{3}(x^2$ + 8$$)^{3/2}$$ + C$$

Question 6

A factory's production rate is given by $$f(x) = x^2$ + 1$$ units per hour. Using a left Riemann sum with 10 subintervals on $$[5, 10]$$, what is the approximate total production?

Accepted Answer

$$278.125$$

Question 7

Evaluate the integral $$\int x e$$^{x^2}$$ \, dx.$$

Accepted Answer

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

Question 8

Find the net area of the region bounded by $$f(x) = x^2$ - 25$$ and the $$x-$$axis on $$[2, 4].$$

Accepted Answer

$$-\frac{94}{3}$$

Question 9

A chemical reaction's rate is modeled by $$f(x) = x(x+1)(x-1). $$What is the net area under the curve from $$x = -1 to$$ $$x = 2$$?

Accepted Answer

$$\frac{9}{4}$$

Question 10

Which of the following is the correct antiderivative of $$\int 2x(x^2$ - 1$$)^{99}$$ \, dx$$?

Accepted Answer

$$\frac{(x^2$ - 1$$)^{100}$$}{100} + C$$

Question 11

A company wants to estimate the area under the curve $$f(x) = 8 - 2x$$ from $$x = 0 to$$ $$x = 6$$ using geometry. What is the total area and the net area?

Accepted Answer

Area: $$20$$, Net Area: $$12$$

Question 12

Evaluate the integral $$\int \sin$$^{10}$$(x) \cos(x) \, dx.$$

Accepted Answer

$$\frac{1}{11}\sin$$^{11}$$(x) + C$$

Question 13

Which of the following is the correct antiderivative of $$\int 5x(12x^3$ - 10x) \, dx$$?

Accepted Answer

$$12x^5$$ - \frac{50}{3}$$x^3 + C$

Question 14

A biologist models the growth of a population with $$f(x) = x^2$ + 1 on$$ $$[5, 10]. $$Using a midpoint Riemann sum with 10 subintervals, what is the approximate area under the curve?

Accepted Answer

$$296.5625$$

Question 15

Evaluate the definite integral $$\int_{3}^{5} \frac{2x}{(x^2$ + 1)^2$} \, dx.$$

Accepted Answer

$$\frac{4}{65}$$

Question 16

Which of the following is the correct antiderivative of $$\int \frac{5x}{x^2$ + 1} \, dx$$?

Accepted Answer

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

Question 17

A physics experiment measures the area under $$f(x) = x^2$ + 1$$ from $$x = 5 to$$ $$x = 10$$ using a right Riemann sum with 10 subintervals. What is the approximate area?

Accepted Answer

$$315.625$$

Question 18

Evaluate the integral $$\int \frac{4x^4$ - 6x^2$}{x} \, dx.$$

Accepted Answer

$$x^4$$ - $$3x^2 + C$

Question 19

Which of the following is the correct antiderivative of $$\int \frac{2x^2$}{\sqrt{1 - 4x^3$}} \, dx$$?

Accepted Answer

$$-\frac{1}{3}\sqrt{1 - 4x^3$} + C$$

Question 20

Evaluate the integral $$\int_{0}^{1} 2x(4 - x^2$) \, dx.$$

Accepted Answer

$$-42$$

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