Q: Suppose you are asked to evaluate $$\int \frac{dx}{(1 + x^2$)^2$$}$ using trigonometric substitution. What is the resulting antiderivative?

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

Q: Which of the following is the correct evaluation for $$\int \frac{2 - \sin 2\theta}{\cos^2$ 2\theta} d\theta$$?

$$\tan 2\theta - \frac{1}{2} \sec^2$ 2\theta + C$$

Question 1

Evaluate the integral $$\int \frac{dx}{\sqrt{-x^2$ - 8x - 7}} by $$completing the square. Which of the following is the correct antiderivative?

Accepted Answer

$$\sin$$^{-1}$$ \left( \frac{x+4}{3} \right) + C$$

Question 2

Which trigonometric identity is most useful for evaluating $$\int \frac{2 - \sin 2	heta}{\cos^2$ 2	heta} d	heta$$?

Accepted Answer

$$\frac{1}{\cos^2$ x} = \sec^2$ x$$

Question 3

Evaluate $$\int_{\ln 2}^{-1} 3t e^t dt$$ using integration by parts.

Accepted Answer

$$-\frac{3}{2}(\ln 2 + 1)$$

Question 4

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

Accepted Answer

$$x = 	an 	heta$$

Question 5

Find $$\int \frac{8x + 5}{2x^2$ + 3x + 1} dx$$ using partial fractions. Which of the following is the correct answer?

Accepted Answer

$$\ln|(2x+1)(x+1)^3$| + C$$

Question 6

Evaluate the improper integral $$\int_{-\infty}^{-1} \frac{dx}{(x-1)^4$}. $$What is the value?

Accepted Answer

$$\frac{1}{24}$$

Question 7

Evaluate the improper integral $$\int_{-\infty}^{\infty} \frac{dx}{x^2$ + 100}. $$What is the value?

Accepted Answer

$$\frac{\pi}{10}$$

Question 8

Evaluate $$\$$int_1$$^2 \frac{dx}{\sqrt{x-1}}. $$What is the value?

Accepted Answer

$$2$$

Question 9

A pharmaceutical company needs to calculate the area under the curve $$y = \frac{1}{x^2$ + 100}$$ from $$x = -\infty to$$ $$x = \infty to $$determine the total drug concentration over time. What is the total area?

Accepted Answer

$$\frac{\pi}{10}$$

Question 10

Which of the following integrals requires the use of completing the square before applying a trigonometric substitution?

Accepted Answer

$$\int \frac{dx}{\sqrt{-x^2$ - 8x - 7}}$$

Question 11

Suppose you are asked to evaluate $$\int \frac{dx}{(1 + x^2$)^2$$}$ using trigonometric substitution. What is the resulting antiderivative?

Accepted Answer

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

Question 12

A physicist is modeling the decay of a radioactive substance and needs to evaluate $$\int_{-\infty}^{-1} \frac{dx}{(x-1)^4$}. $$What is the result?

Accepted Answer

$$\frac{1}{24}$$

Question 13

Which of the following is the correct evaluation for $$\int \frac{2 - \sin 2	heta}{\cos^2$ 2	heta} d	heta$$?

Accepted Answer

$$	an 2	heta - \frac{1}{2} \sec^2$ 2	heta + C$$

Question 14

Which of the following best describes the process for evaluating $$\int_{\ln 2}^{-1} 3t e^t dt$$?

Accepted Answer

Use integration by parts with $$u = 3t$$ and $$dv = e^t dt$$

Question 15

A medical researcher is analyzing the rate of absorption of a drug, modeled by $$f(x) = \frac{1}{\sqrt{x-1}}$$ for $$x \geq 1. $$What is the total absorption from $$x=1 to$$ $$x=2$$?

Accepted Answer

$$2$$

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