Q: Find the area enclosed by one leaf of the four-leaved rose $$r = \sin(2\theta).$$

$$\frac{1}{2} \$$int_0$$^{\pi/2} \sin^2$(2\theta) d\theta$$

Question 1

Evaluate the integral $$\int x^2$ e$$^{-2x}$$ dx$$ using integration by parts.

Accepted Answer

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

Question 2

Which of the following is the correct partial fraction decomposition form for $$f(x) = \frac{4x^3$ + 3x^2$ + 2x + 1}{(x^2$ + x + 1)^2$ (x + 5)^4$}$$?

Accepted Answer

$$\frac{A}{x + 5} + \frac{B}{(x + 5)^2$} + \frac{C}{(x + 5)^3$} + \frac{D}{(x + 5)^4$} + \frac{Ex + F}{x^2$ + x + 1} + \frac{Gx + H}{(x^2$ + x + 1)^2$}$$

Question 3

Given the parametric equations $$x = 3t$$, $$y = 9t^2$, what is the equation of the tangent line at t = 1$?

Accepted Answer

$$y = 6x - 9$$

Question 4

Convert the polar equation $$r \sin 	heta + r \cos 	heta = 1 to $$Cartesian form.

Accepted Answer

$$x + y = 1$$

Question 5

Find the area enclosed by the cardioid $$r = 1 + 2 \sin 	heta.$$

Accepted Answer

$$3\pi + 4$$

Question 6

Which of the following is the correct polar form of the Cartesian equation $$x^2 + (y$$ - $$2)^2 = 4$?

Accepted Answer

$$r^2 - 4r$$ \sin 	heta = 4$$

Question 7

Evaluate $$\int \frac{dx}{x^2$ \sqrt{x^2$ - 1}}.$$

Accepted Answer

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

Question 8

Find the value of $$a$$ that makes $$f(x) = \begin{cases} a e$$^{-x}$$ & x \geq 0 \ 0 & x \u003c 0 \end{cases} a $$probability density function.

Accepted Answer

$$a = 1$$

Question 9

Given $$r = \$$theta^2$$, 0$$ \leq 	heta \leq \sqrt{5}$$, what is the formula for the arc length of this curve?

Accepted Answer

$$\$$int_0$$^{\sqrt{5}} \sqrt{\$$theta^4$$ + 4\$$theta^2$$ + 1} d	heta$$

Question 10

Find the eccentricity of the conic section $$2x^2$$ + $$y^2 = 2$.

Accepted Answer

$$e = \sqrt{1 - \frac{1}{2}} = \frac{1}{\sqrt{2}}$$

Question 11

Evaluate $$\int_{0}^{\pi/4} x^2$ \sin x dx$$ using integration by parts.

Accepted Answer

$$\left[ -x^2$ \cos x + 2x \sin x + 2 \cos x \right]_0$$^{\pi/4}$$

Question 12

Which of the following integrals would you use a trigonometric substitution $$x = 7 \sin 	heta to $$evaluate?

Accepted Answer

$$\int \sqrt{49 - x^2$} dx$$

Question 13

Given the parametric equations $$x = \sin t$$, $$y = \cos 2t$$, what is $$\frac{dy}{dx} in $$terms of $$t$$?

Accepted Answer

$$\frac{-2 \sin 2t}{\cos t}$$

Question 14

Find the area enclosed by one leaf of the four-leaved rose $$r = \sin(2	heta).$$

Accepted Answer

$$\frac{1}{2} \$$int_0$$^{\pi/2} \sin^2$(2	heta) d	heta$$

Question 15

Which of the following is the correct partial fraction decomposition form for $$f(x) = \frac{3x^2$ - 5x - 1}{(x^2$ + 2)^2$}$$?

Accepted Answer

$$\frac{Ax + B}{x^2$ + 2} + \frac{Cx + D}{(x^2$ + 2)^2$}$$

Question 16

Evaluate $$\int \frac{5x - 13}{x^2$ - 5x + 6} dx.$$

Accepted Answer

$$5 \ln|x - 2| - 13 \ln|x - 3| + C$$

Step-by-Step Calculus Practice Exam 2 Guidance

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