Math 161 — Test 2 Practice Questions

Problem 1. Evaluate the following integrals using a trigonometric substitution. No credit for any other methods.

1. (a)
   \(\displaystyle\int \frac{x^3}{\sqrt{x^2+25}}\, dx\)
2. (b)
   \(\displaystyle\int \frac{x}{\sqrt{4-x^2}}\, dx\)
3. (c)
   \(\displaystyle\int \frac{\sqrt{x^2-4}}{x}\, dx\)
4. (d)
   \(\displaystyle\int \frac{x^2}{\sqrt{9-x^2}}\, dx\)
5. (e)
   \(\displaystyle\int \frac{1}{\sqrt{4+x^2}}\, dx\)
6. (f)
   \(\displaystyle\int \frac{x}{\sqrt{x^2-9}}\, dx\)
7. (g)
   \(\displaystyle\int_0^5 \frac{dt}{\sqrt{25+t^2}}\)

Problem 2. Evaluate the following integrals using partial fraction decomposition.

1. (a)
   \(\displaystyle\int \frac{2x+19}{(x-3)(x+2)}\, dx\)
2. (b)
   \(\displaystyle\int \frac{10x+2}{(3x+1)(x-1)}\, dx\)
3. (c)
   \(\displaystyle\int_0^1 \frac{x-6}{x^2-6x+8}\, dx\)
4. (d)
   \(\displaystyle\int \frac{5x^2+2x-5}{x^3-x}\, dx\)
5. (e)
   \(\displaystyle\int \frac{x+2}{x^3-2x^2}\, dx\)
6. (f)
   \(\displaystyle\int \frac{x^2+1}{(x-2)(x-5)^2}\, dx\)
7. (g)
   \(\displaystyle\int \frac{2x}{(x+1)(x^2+1)}\, dx\)
8. (h)
   \(\displaystyle\int_1^5 \frac{x^2+5}{6x-x^2}\, dx\)
9. (i)
   \(\displaystyle\int_1^2 \frac{4y^2-7y-12}{y(y+2)(y-3)}\, dy\)
10. (j)
    \(\displaystyle\int_3^4 \frac{2x^2-4}{x^3-2x^2}\, dx\)

Problem 3. Determine whether each of the following improper integrals converges or diverges (show your work). If it converges, determine its value.

1. (a)
   \(\displaystyle\int_5^{\infty} \frac{4}{(x-2)^3}\, dx\)
2. (b)
   \(\displaystyle\int_{0}^{\infty} \frac{1}{\sqrt[3]{x+2}}\, dx\)
3. (c)
   \(\displaystyle\int_{3}^{\infty} 12e^{-2x}\, dx\)
4. (d)
   \(\displaystyle\int_0^{\infty} \frac{1}{(2x+5)^3}\, dx\)
5. (e)
   \(\displaystyle\int_0^{\infty} \frac{1}{1+x^2}\, dx\)
6. (f)
   \(\displaystyle\int_0^{\infty} \frac{x^2}{5+x^3}\, dx\)
7. (g)
   \(\displaystyle\int_0^{\infty} \frac{x^2}{\sqrt{6+x^3}}\, dx\)
8. (h)
   \(\displaystyle\int_2^{\infty} \frac{e^{-1/x}}{x^2}\, dx\)
9. (i)
   \(\displaystyle\int_1^{4} \frac{1}{\sqrt{x-1}}\, dx\)

Problem 4. Find the arc length of the curve \(y = 3x - 5\) on the interval \(1 \leq x \leq 3\).

Problem 5. Find the arc length of the curve \(y = 4x^{3/2} + 5\) on the interval \(0 \leq x \leq 1\).

Problem 6. Find the arc length of the curve \(y = \ln(\sec x)\) on the interval \(0 \leq x \leq \dfrac{\pi}{4}\).

Problem 7. Find general solutions of the following differential equations. Write your solution as \(y = f(x)\).

1. (a)
   \(\dfrac{dy}{dx} = \dfrac{1}{xy}\)
2. (b)
   \(\dfrac{dy}{dx} = \dfrac{1}{1+x^2}\)
3. (c)
   \(\dfrac{dy}{dx} = ye^x\)
4. (d)
   \(e^{-2x}\,\dfrac{dy}{dx} = \dfrac{1}{y}\)

Problem 8. Find the solution of the differential equation that satisfies the given initial condition. Write your solution as \(y = f(x)\).

1. (a)
   \(\dfrac{dy}{dx} = 4x, \quad y(1) = 5\)
2. (b)
   \(\dfrac{dy}{dx} = \dfrac{1}{\sec^2 y}, \quad y(2) = 0\)
3. (c)
   \(xy^2\,\dfrac{dy}{dx} = x+1, \quad y(1) = 3\)
4. (d)
   \(\dfrac{dy}{dx} = y^2 e^x, \quad y(0) = 1\)
5. (e)
   \(\dfrac{dy}{dx} = \dfrac{6x^2}{y}, \quad y(0) = -2\)
6. (f)
   \(\dfrac{dy}{dx} = \dfrac{\sin(x)}{2y}, \quad y(0) = -1\)

Problem 9. Determine whether the following sequences converge or diverge. If it converges, determine its limit.

1. (a)
   \(\displaystyle a_n = \frac{5n^2+2n+20}{3n^5-3n^2-5}\)
2. (b)
   \(\displaystyle a_n = \frac{3n^4-2n}{8n^4+100n^2+1000}\)
3. (c)
   \(\displaystyle a_n = \frac{2n^3+1}{1500n^2+100n+3000}\)
4. (d)
   \(\displaystyle a_n = 2^{-n}\)
5. (e)
   \(\displaystyle a_n = 3^n \cdot 2^{1-n}\)
6. (f)
   \(\displaystyle a_n = 4^n\)
7. (g)
   \(\displaystyle a_n = 2^{\frac{2n}{n+1}}\)
8. (h)
   \(\displaystyle a_n = 3^{\frac{n}{n^2+1}}\)
9. (i)
   \(\left\{-4,\ 12,\ -36,\ 108,\ -324,\ \ldots\right\}\)