Q: Which of the following is the Cauchy principal value for $$\displaystyle \int_{-1}^1 \frac{dx}{x}$$?

$$\displaystyle \lim_{a \to 0^+} \left( \int_{-1}^{-a} \frac{dx}{x} + \int_{a}^1 \frac{dx}{x} \right)$$

Question 1

Which of the following is the correct partial fraction decomposition for $$\displaystyle \frac{2x^2$-3}{x(x^2$-2x+1)}$$?

Accepted Answer

$$\displaystyle \frac{A}{x} + \frac{B}{x-1} + \frac{C}{(x-1)^2$}$$

Question 2

Evaluate $$\displaystyle \$$int_1$$^9 \sqrt{1 + \left( \frac{dx}{dy} \$$right)^2$$} dy$$ where $$x = \frac{1}{2} \sqrt{y-3}.$$

Accepted Answer

$$\displaystyle \frac{32}{3}$$

Question 3

Which substitution is most appropriate for evaluating $$\displaystyle \int \frac{1}{x^2$ \sqrt{25-x^2$}} dx$$?

Accepted Answer

$$x = 5 \sin 	heta$$

Question 4

Given $$\displaystyle \int \sin^2$ x \sqrt{\cos x} dx$$, which substitution simplifies the integral?

Accepted Answer

Let $$u = \cos x$$

Question 5

Evaluate $$\displaystyle \$$int_1$$^3 \cos(\ln x) dx.$$

Accepted Answer

$$\displaystyle \frac{1}{2} \left[ 3 \cos(\ln 3) + 3 \sin(\ln 3) - 1 \right]$$

Question 6

The Trapezoidal Rule will give an underestimate for $$\displaystyle \int_a^b f(x) dx if$$ $$f(x) is$$:

Accepted Answer

Continuous and decreasing on $$[a, b]$$

Question 7

Which of the following integrals represents the surface area obtained by rotating $$y = \sec x$$, $$0 \leq x \leq \frac{\pi}{4}$$ about the $$y-$$axis?

Accepted Answer

$$\displaystyle \$$int_0$$^{\frac{\pi}{4}} 2\pi \sec x \sqrt{1 + (\sec x 	an x)^2$} dx$$

Question 8

For the sequence $$a_n = \frac{(-1$$)^{n+1}$$}{n^2$}$$, what is $$\lim_{n 	o \infty} a_n$$?

Accepted Answer

$$0$$

Question 9

Consider the sequence $$a_n = \frac{n}{2n+1}$$ for odd $$n$$ and $$a_n = \frac{-n}{2n+1}$$ for even $$n. $$What can be said about its convergence?

Accepted Answer

The sequence diverges because the odd and even terms have different limits.

Question 10

Which of the following is the derivative of $$\arctan x$$?

Accepted Answer

$$\displaystyle \frac{1}{1+x^2$}$$

Question 11

For $$f(x) = \frac{\arctan x}{1+x^2$}$$, what is the sign of $$f'(x)$$ for $$x \geq 1$$?

Accepted Answer

Negative

Question 12

Which test can be used to determine the convergence of $$\displaystyle \sum_{n=1}^{\infty} \frac{\arctan n}{1+n^2$}$$?

Accepted Answer

Integral Test

Question 13

Evaluate $$\displaystyle \int \frac{1}{x^2$ \sqrt{25-x^2$}} dx$$ using trigonometric substitution.

Accepted Answer

$$\displaystyle \frac{1}{25} \left( \frac{\sqrt{25-x^2$}}{x} \right) + C$$

Question 14

Find $$\displaystyle \int \sin^2$ x \sqrt{\cos x} dx.$$

Accepted Answer

$$\displaystyle \frac{2}{7} (\sqrt{\cos x})^2$ - \frac{2}{3} (\sqrt{\cos x})^3$ + C$$

Question 15

Given $$\displaystyle \$$int_2$$^3 \left( \frac{x}{x} + \frac{5}{x-1} - \frac{1}{(x-1)^2$} \right) dx$$, what is the value?

Accepted Answer

$$\displaystyle 3 \ln 3 + 2 \ln 2 - \frac{1}{2}$$

Question 16

If $$\lim_{n 	o \infty} |a_n| = 2$$, what can be said about the convergence of $$\sum_{n=1}^{\infty} a_n$$?

Accepted Answer

The series diverges.

Question 17

Which of the following is the Cauchy principal value for $$\displaystyle \int_{-1}^1 \frac{dx}{x}$$?

Accepted Answer

$$\displaystyle \lim_{a 	o 0^+} \left( \int_{-1}^{-a} \frac{dx}{x} + \int_{a}^1 \frac{dx}{x} ight)$$

Question 18

Which of the following statements about the function $$f(x) = \arctan x is $$true?

Accepted Answer

$$f(x) is $$continuous and positive for all $$x \geq 0$$

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