Question 1

Evaluate the integral $$\int \frac{3\, dx}{\sqrt{1 + 9x^2$}}$$ using an appropriate substitution.

Accepted Answer

$$\$$sinh^{-1}$$(3x) = \ln\left|3x + \sqrt{1 + 9x^2$}\right| + C$$

Question 2

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

Accepted Answer

$$x = 2\sin	heta$$

Question 3

Find the area of the region enclosed by $$y = x \cos x$$ and the $$x-$$axis for $$\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}.$$

Accepted Answer

$$\left|\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} x \cos x\, dx\right|$$

Question 4

Evaluate $$\int \frac{2x + 1}{x^2$ - 7x + 12} dx$$ using partial fractions.

Accepted Answer

$$\int \frac{2x + 1}{x^2$ - 7x + 12} dx = \int \left(\frac{3}{x-3} - \frac{1}{x-4}\right) dx = 3\ln|x-3| - \ln|x-4| + C$$

Question 5

Which of the following is the correct formula for Simpson's Rule for approximating $$\int_a^b f(x) dx$$ with $$n$$ even subintervals?

Accepted Answer

$$S = \frac{\Delta x}{3} [f(x_0$) + 4f(x_1$) + 2f(x_2$) + 4f(x_3$) + \cdots + 4f(x_{n-1}) + f(x_n)]$$

Question 6

Evaluate $$\int_{0}^{2} \frac{dx}{8 + 2x^2$}.$$

Accepted Answer

$$\frac{1}{4} \arctan(x/2) \Big|_0^2$

Question 7

Which of the following integrals is improper of Type II?

Accepted Answer

$$\int_{0}^{1} \frac{dx}{\sqrt{1-x}}$$

Question 8

Find the average value of $$y = \ln x on $$the interval $$[1, e].$$

Accepted Answer

$$\frac{1}{e-1} \int_{1}^{e} \ln x dx$$

Question 9

Which identity is used to evaluate $$\int \sin mx \cos nx dx$$?

Accepted Answer

$$\frac{1}{2}[\sin(m-n)x + \sin(m+n)x]$$

Question 10

Evaluate $$\int \frac{8x^2$ + 8x + 2}{(4x^2$ + 1)^2$} dx$$ using partial fractions.

Accepted Answer

$$2 \arctan(2x) + \frac{2x}{4x^2$ + 1} + C$$

Question 11

Which of the following is the correct setup for the volume generated by revolving $$y = x \sin x$$, $$0 \leq x \leq \pi$$, about the $$y-$$axis?

Accepted Answer

$$V = 2\pi \int_{0}^{\pi} (x \sin x) x dx$$

Question 12

Evaluate $$\int \frac{2 dx}{x^3$ \sqrt{x^2$ - 1}}$$ for $$x \u003e 1$$ using trigonometric substitution.

Accepted Answer

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

Question 13

Which of the following best describes the error bound for the Trapezoidal Rule?

Accepted Answer

$$|E_T| \leq \frac{M(b-a)^3$}{12n^2$}$$, where $$M is an $$upper bound for $$|f''(x)| on$$ $$[a,b]$$

Question 14

Evaluate $$\int_{1}^{e} \frac{dy}{y \sqrt{1 + (\ln y)^2$}}$$ using substitution.

Accepted Answer

$$\arctan(\ln y) \Big|_{1}^{e}$$

Question 15

Which of the following is the correct partial fraction decomposition for $$\frac{x^4$}{x^2$ - 1}$$?

Accepted Answer

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

Question 16

Given the region bounded by $$y = 	an$$^{-1}$$ x$$, $$y = 0$$, and $$x = 1$$, what is the area of the region?

Accepted Answer

$$\int_{0}^{1} 	an$$^{-1}$$ x dx$$

Question 17

Which of the following is the correct trigonometric substitution for $$\int \frac{dy}{y^3$ \sqrt{y^2$ - 25}}$$, $$y \u003e 5$$?

Accepted Answer

$$y = 5 \sec 	heta$$

Step-by-Step Calculus Guidance: Integration, Area, Volume, Trigonometric Substitution, Partial Fractions, Numerical Methods, and Improper Integrals

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