Question 1

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

Accepted Answer

$$\arctan(x) + C$$

Question 2

A biologist models the population of a bacteria culture by $$P(t) = 100e$$^{0.03t}$$, where t$ is in hours. What is $$\$$int_0$$^{24} P(t)\,dt$$ and what does it represent?

Accepted Answer

$$\frac{100}{0.03}(e$$^{0.72}$$ - 1)$$; it represents the total bacteria-hours over 24 hours.

Question 3

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

Accepted Answer

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

Question 4

Which substitution is most appropriate for evaluating $$\int \frac{1}{1+4x^2$} dx$$?

Accepted Answer

$$u = 2x$$

Question 5

Given $$f(x) = \sin(3x)$$, what is $$\int f(x) dx$$?

Accepted Answer

$$-\frac{1}{3}\cos(3x) + C$$

Question 6

A town's energy consumption rate is modeled by $$r(t) = 4 + 8\sin(0.262t + 4.7) + \cos(0.262t + 9.4). $$What does $$\$$int_0$$^{24} r(t) dt$$ represent?

Accepted Answer

The total energy consumed in a 24-hour period, in units of megawatts-hours.

Question 7

Evaluate $$\int \sec^2$ x dx.$$

Accepted Answer

$$	an x + C$$

Question 8

Which of the following integrals is best solved using integration by parts?

Accepted Answer

$$\int x \ln(x) dx$$

Question 9

Find the general antiderivative of $$f(x) = \cos(x^2$).$$

Accepted Answer

It cannot be expressed in terms of elementary functions.

Question 10

Evaluate $$\$$int_0$$^1 \ln(x) dx.$$

Accepted Answer

$$-1$$

Question 11

Which of the following is the correct partial fraction decomposition for $$\frac{2x-3}{(x-3)(x-4)}$$?

Accepted Answer

$$\frac{A}{x-3} + \frac{B}{x-4}$$

Question 12

Evaluate $$\int \frac{1}{\sqrt{1-x^2$}} dx.$$

Accepted Answer

$$\arcsin(x) + C$$

Question 13

Which of the following best describes the graph of $$y = \ln(x)$$?

Accepted Answer

It passes through $$(1,0)$$ and increases without bound as $$x 	o \infty.$$

Question 14

Which of the following integrals can be evaluated immediately using a basic antiderivative?

Accepted Answer

$$\int \sin(x) dx$$

Question 15

Evaluate $$\int e$$^{4t}$$ dt.$$

Accepted Answer

$$\frac{1}{4}e$$^{4t}$$ + C$$

Question 16

Which of the following is the correct use of the linearity property of integrals?

Accepted Answer

$$\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$$

Question 17

A planet has density $$\rho(r) = \$$rho_0$$ e$$^{-kr}$$, r$ is the distance from the center. The total mass is M$$ = \$$int_0^R 4$$\pi $$r^2$$ \rho(r) dr$$. Which method is most appropriate to evaluate M$?

Accepted Answer

Substitution $$u = -kr$$

Question 18

Evaluate $$\int 	an(x) dx.$$

Accepted Answer

$$-\ln|\cos(x)| + C$$

Question 19

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

Accepted Answer

$$\ln|x| + C$$

Calculus II Study Notes: Integration Techniques, Antiderivatives, and Applications

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