Q: Evaluate the integral $$\int \frac{x-1}{x^2$+3} \, dx.$$

$$\frac{1}{2} \ln(x^2$+3) - \frac{1}{\sqrt{3}} \tan$$^{-1}$$\left(\frac{x}{\sqrt{3}}\right) + C$$

Q: Which trigonometric identity is used to simplify $$\sin^2$(2t)$$ before integrating?

$$\sin^2$(\theta) = \frac{1}{2}(1 - \cos(2\theta))$$

Q: Which of the following is the correct antiderivative of $$\frac{1}{x^2$+3}$$?

$$\frac{1}{\sqrt{3}} \tan$$^{-1}$$\left(\frac{x}{\sqrt{3}}\right) + C$$

Question 1

Evaluate the integral $$\int \frac{x-1}{x^2$+3} \, dx.$$

Accepted Answer

$$\frac{1}{2} \ln(x^2$+3) - \frac{1}{\sqrt{3}} 	an$$^{-1}$$\left(\frac{x}{\sqrt{3}}ight) + C$$

Question 2

Which substitution is most appropriate for evaluating $$\int \frac{x}{x^2$+3} \, dx$$?

Accepted Answer

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

Question 3

Given $$	an 	heta = \frac{x}{\sqrt{3}}$$, what is $$\sec 	heta in $$terms of $$x$$?

Accepted Answer

$$\frac{\sqrt{x^2$+3}}{\sqrt{3}}$$

Question 4

Evaluate $$\int 	an^2$ 	heta \, d	heta.$$

Accepted Answer

Question 5

Which of the following is the correct result for $$\int x e$$^{2x}$$ \, dx$$?

Accepted Answer

$$\frac{1}{2} x e$$^{2x}$$ - \frac{1}{4} e$$^{2x}$$ + C$$

Question 6

In the integration by parts formula $$\int u \, dv = uv - \int v \, du$$, which choices of $$u$$ and $$dv$$ are most appropriate for $$\int x e$$^{2x}$$ \, dx$$?

Accepted Answer

$$u = x$$, $$dv = e$$^{2x}$$ dx$$

Question 7

Evaluate $$\int \sin^2$(2t) \, dt.$$

Accepted Answer

$$-\frac{1}{2} \cos(2t) + \frac{1}{6} \cos^3$(2t) + C$$

Question 8

Which trigonometric identity is used to simplify $$\sin^2$(2t)$$ before integrating?

Accepted Answer

$$\sin^2$(	heta) = \frac{1}{2}(1 - \cos(2	heta))$$

Question 9

Suppose you are modeling the cooling of a hot object in a room using Newton's Law of Cooling, which involves an exponential function. Which integral would you use to find the total heat lost over time if the rate is proportional to $$e^{-kt}$$?

Accepted Answer

$$\int e$$^{-kt}$$ \, dt$$

Question 10

A pharmaceutical company is studying the concentration of a drug in the bloodstream, modeled by $$C(t) = C_0$ e$$^{-kt}$$. What is the total amount of drug in the bloodstream from t=0$ to t$$=\infty$$?

Accepted Answer

$$\frac{C_0$}{k}$$

Question 11

Which of the following integrals would be most appropriate for finding the area under the curve $$y = \frac{1}{x^2$+3}$$ from $$x=0 to$$ $$x=1$$?

Accepted Answer

$$\$$int_0$$^1 \frac{1}{x^2$+3} \, dx$$

Question 12

If $$u = x^2$ + 3$$, what is $$du$$?

Accepted Answer

$$2x \, dx$$

Question 13

Which of the following is the correct antiderivative of $$\frac{1}{x^2$+3}$$?

Accepted Answer

$$\frac{1}{\sqrt{3}} 	an$$^{-1}$$\left(\frac{x}{\sqrt{3}}ight) + C$$

Question 14

A triangle is drawn with sides $$x$$, $$\sqrt{3}$$, and $$\sqrt{x^2$+3}. If$$ $$	an 	heta = \frac{x}{\sqrt{3}}$$, what is the length of the hypotenuse?

Accepted Answer

$$\sqrt{x^2$+3}$$

Question 15

Which of the following best describes the method used to solve $$\int \frac{x-1}{x^2$+3} \, dx$$?

Accepted Answer

Splitting the integrand and using substitution and trigonometric integration

Question 16

Which of the following is the correct result for $$\int \sec^2$ 	heta \, d	heta$$?

Accepted Answer

$$	an 	heta + C$$

Question 17

If $$\int f(x) \, dx = F(x) + C$$, what is $$\frac{d}{dx} F(x)$$?

Accepted Answer

$$f(x)$$

Question 18

Which of the following is a correct application of integration by parts for $$\int x e$$^{2x}$$ \, dx$$?

Accepted Answer

$$u = x$$, $$dv = e$$^{2x}$$ dx$$, $$du = dx$$, $$v = \frac{1}{2} e$$^{2x}$$

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