Q: Which of the following limits requires the use of logarithms and L'Hôpital's Rule to evaluate?

$$\lim_{x \to \infty} \left( \frac{x}{x+1} \right)^x$$

Q: Which substitution is most appropriate for evaluating $$\int \frac{x}{\sqrt{x^2$ + 10x + 9}} \, dx$$?

Complete the square and use $$x + 5 = 2 \tan \theta$$

Question 1

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

Accepted Answer

$$\int x^2$ \sin x \, dx$$

Question 2

Evaluate $$\$$int_0$$^{\frac{\pi}{4}} x^3$ \cos(2x) \, dx$$ using integration by parts. Which function should you choose for $$u$$ according to the LIATE rule?

Accepted Answer

$$u = x^3$

Question 3

Which trigonometric substitution is appropriate for evaluating $$\int \sqrt{16 - x^2$} \, dx$$?

Accepted Answer

$$x = 4 \sin 	heta$$

Question 4

Given $$\int \frac{x}{x^2$ + 10x + 9} \, dx$$, which method should be used to evaluate the integral?

Accepted Answer

Partial fraction decomposition

Question 5

Which of the following is an indeterminate form that can be evaluated using L'Hôpital's Rule?

Accepted Answer

$$\frac{0}{0}$$

Question 6

Evaluate $$\lim_{x 	o 0^+} x$$^{1/2}$$ \ln x.$$

Accepted Answer

$$0$$

Question 7

Which of the following integrals is improper due to an infinite interval?

Accepted Answer

$$\$$int_0$$^{\infty} \frac{1}{(x+2)^2$} \, dx$$

Question 8

Which trigonometric identity is used to simplify $$\int \sin^3$ x \cos^2$ x \, dx$$?

Accepted Answer

$$\sin^2$ x = 1 - \cos^2$ x$$

Question 9

Given $$\int \frac{dx}{x \sqrt{x^2$ + 1}}$$, which substitution is most appropriate?

Accepted Answer

$$x = 	an 	heta$$

Question 10

Evaluate $$\$$int_0$$^{\pi/2} \cot 	heta \, d	heta. $$What is the result?

Accepted Answer

The integral diverges

Question 11

Which of the following is the general solution to the differential equation $$y'(t) = 5 \cos t + 6 \sin 3t$$?

Accepted Answer

$$y(t) = 5 \sin t - 2 \cos 3t + C$$

Question 12

For the integral $$\int \frac{1}{x^2$ + 6x + 13} \, dx$$, what is the first step to solve it?

Accepted Answer

Complete the square in the denominator

Question 13

Which of the following limits requires the use of logarithms and L'Hôpital's Rule to evaluate?

Accepted Answer

$$\lim_{x 	o \infty} \left( \frac{x}{x+1} ight)^x$$

Question 14

Given $$\int \frac{x^2$ + x + 2}{(x+1)(x^2$ + 1)} \, dx$$, which numerator form should be used for the partial fraction decomposition?

Accepted Answer

$$\frac{A}{x+1} + \frac{Bx + C}{x^2$ + 1}$$

Question 15

Which of the following integrals converges?

Accepted Answer

$$\int_{-\infty}^0 \frac{dx}{x^2$ + 1}$$

Question 16

Solve the initial value problem $$y'(t) = 10 e$$^{-t/2}$$, y(0) = 4$. What is y(t$$)$$?

Accepted Answer

$$y(t) = -20 e$$^{-t/2}$$ + 24$$

Question 17

Which of the following is the correct partial fraction decomposition for $$\frac{x}{x^2$ + 10x + 9}$$?

Accepted Answer

$$\frac{A}{x+1} + \frac{B}{x+9}$$

Question 18

Which substitution is most appropriate for evaluating $$\int \frac{x}{\sqrt{x^2$ + 10x + 9}} \, dx$$?

Accepted Answer

Complete the square and use $$x + 5 = 2 	an 	heta$$

Question 19

Which of the following is the correct general solution to $$y''(t) = 10t^3$ - 144t^7$ + 12t$$?

Accepted Answer

$$y(t) = \frac{5}{2} t^4$ - 16 t^8$ + 6 t^2$ + C_1$ t + C_2$

exam 2

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