Question 1

Evaluate the integral: $$\int x^2$ e$$^{3x}$$ \, dx$$

Accepted Answer

$$\frac{1}{3}x^2$ e$$^{3x}$$ - \frac{2}{9}x e$$^{3x}$$ + \frac{2}{27}e$$^{3x}$$ + C$$

Question 2

Evaluate the definite integral: $$\$$int_0$$^1 x \cos(x^2$) \, dx$$

Accepted Answer

$$\frac{1}{2}\sin(1)$$

Question 3

Evaluate the integral: $$\int \frac{1}{x^2$+4} \, dx$$

Accepted Answer

$$\frac{1}{2}\arctan\left(\frac{x}{2}\right) + C$$

Question 4

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

Accepted Answer

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

Question 5

Set up the integral to find the area between $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$.

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) \, dx$$

Question 6

Express the integral $$\int \frac{2x+1}{x^2$+x} \, dx as a $$sum of partial fractions and evaluate.

Accepted Answer

$$2\ln|x| - \ln|x+1| + C$$

Question 7

Use the Trapezoidal Rule with $$n=4$$ steps to estimate $$\$$int_0$$^2 x^2$ \, dx.$$

Accepted Answer

$$2.75$$

Question 8

Use Simpson's Rule with $$n=4 to $$estimate $$\$$int_0$$^2 x^2$ \, dx.$$

Accepted Answer

$$2.6667$$

Question 9

Determine whether the improper integral $$\$$int_1$$^\infty \frac{1}{x^2$} \, dx$$ converges or diverges.

Accepted Answer

It converges to $$1$$

Question 10

Use Euler's method with $$\Delta x = 0.2 to $$estimate $$y(0.2)$$ for $$\frac{dy}{dx} = x + y$$, $$y(0) = 1.$$

Accepted Answer

$$1.2$$

Question 11

Solve the differential equation $$\frac{dy}{dx} = 2x$$, $$y(0) = 3.$$

Accepted Answer

$$y = x^2$ + 3$$

Question 12

Solve the initial value problem $$\frac{dy}{dx} = x \cos x$$, $$y(0) = 5.$$

Accepted Answer

Question 13

A 5-kg radioactive substance decays at a rate proportional to its mass. If $$k = 0.12$$ and $$M(0) = 5$$, what is the mass after 10 hours?

Accepted Answer

$$5e^{-1.2} kg$$

Question 14

A 2-kg substance in a 20-kg tank starts with 1-kg of salt. If $$x is $$the amount of salt at time $$t$$, and the rate in is $$0.5 kg/$$min, what is the differential equation for $$x$$?

Accepted Answer

$$\frac{dx}{dt} = 0.5 - \frac{x}{20}$$

Question 15

Identify the equilibrium solutions of $$\frac{dx}{dt} = x^2$ - 4$$ and determine their stability.

Accepted Answer

Equilibria at $$x = -2 ($$stable), $$x = 2 ($$stable)

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