Question 1

Perform polynomial long division to find the remainder $$r(x)$$ when $$x^3 is $$divided by $$x^2 + x - 3$. What is r(x$$)$$?

Accepted Answer

$$r(x) = -4x + 3$$

Question 2

Evaluate the improper integral $$\displaystyle \$$int_0$$^{\infty} \frac{3x^2$}{(x^3$ + 1)^3$} \, dx.$$

Accepted Answer

$$\frac{1}{2}$$

Question 3

Which of the following statements about the integrals $$\displaystyle (i) \int_{-1}^1 \frac{1}{x^2$} dx$$ and $$\displaystyle (ii) \$$int_0$$^1 \frac{1}{e$$^{-x}$$ + e^x} dx is $$correct?

Accepted Answer

(i) is a divergent improper integral; (ii) is not an improper integral.

Question 4

You are tasked with evaluating $$\displaystyle \int \sqrt{x(x-2)} \, dx$$ using a trigonometric substitution. Which triangle is most suitable for setting up the substitution in terms of $$	heta$$?

Accepted Answer

A right triangle with sides $$x-2 ($$opposite), $$x ($$hypotenuse), and $$\sqrt{x^2$ - (x-2)^2$} ($$adjacent).

Question 5

Which of the following is the correct form of the partial fraction decomposition for $$\displaystyle \frac{1}{x^5$ + 4x^3$ + 4x}$$?

Accepted Answer

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

Question 6

Evaluate the definite integral $$\displaystyle \$$int_0$$^1 x^3$ \sqrt{1-x^2$} \, dx.$$

Accepted Answer

$$\frac{2}{15}$$

Question 7

Evaluate $$\displaystyle \int_{-1}^2 \frac{3x-4}{(x-3)(x+2)} \, dx.$$

Accepted Answer

$$\ln(3) - \ln(2)$$

Question 8

Suppose $$\displaystyle \$$int_0$$^{\pi/4} x \sec^2$(x) \, dx = A - \$$int_0$$^{\pi/4} 	an(x) \, dx. $$What is the value of $$A$$?

Accepted Answer

$$\frac{\pi}{4}$$

Question 9

Evaluate $$\displaystyle \int x e^x \, dx.$$

Accepted Answer

$$x e^x + e^x + C$$

Question 10

Evaluate the improper integral $$\displaystyle \$$int_0$$^1 \frac{1}{x^2$ - x} \, dx.$$

Accepted Answer

The integral is divergent.

Question 11

Evaluate $$\displaystyle \$$int_0$$^1 \frac{1}{(x^2$ + 1)^2$} \, dx.$$

Accepted Answer

$$\frac{\pi}{8} + \frac{1}{4}$$

Question 12

Evaluate $$\displaystyle \$$int_0$$^{\pi/2} \cos^3$(x) \, dx.$$

Accepted Answer

$$\frac{2}{3}$$

Question 13

A pharmaceutical company needs to determine the area under the curve $$y = \frac{1}{x^2$}$$ from $$x = 1 to$$ $$x = \infty to $$estimate the total drug concentration over time. What is the value of $$\displaystyle \$$int_1$$^{\infty} \frac{1}{x^2$} dx$$?

Accepted Answer

$$1$$

Question 14

Which substitution is most appropriate for evaluating $$\displaystyle \int \frac{dx}{\sqrt{4 - x^2$}}$$?

Accepted Answer

$$x = 2 \sin(	heta)$$

Question 15

A biologist models the growth of a bacteria population with $$P(t) = 100e$$^{0.05t}$$. What is the total population increase from t=0$ to t=10$ hours?

Accepted Answer

$$100(e^{0.5} - 1$$)$$

Question 16

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

Accepted Answer

$$\displaystyle \$$int_0$$^1 \frac{1}{x} dx$$

Question 17

A chemical reaction's rate is given by $$R(t) = 5e$$^{-2t}$$. What is the total amount of product formed from t=0$ to t$$=\infty$$?

Accepted Answer

$$\frac{5}{2}$$

Question 18

Which trigonometric identity is useful for rewriting $$\cos^3$(x)$$ before integrating?

Accepted Answer

$$\cos^3$(x) = \frac{3\cos(x) + \cos(3x)}{4}$$

Question 19

A medical imaging device uses the function $$f(x) = \frac{1}{x^2$ + 1} to $$model signal intensity. What is the area under the curve from $$x=0 to$$ $$x=1$$?

Accepted Answer

$$\arctan(1) - \arctan(0)$$

Step-by-Step Guidance for MA 16200 Second Midterm (Calculus II Topics)

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