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1.  Compute the indefinite integral   \( \int \frac{1}{x+9}\) dx

a.  (1/3) arctan(x/3) + c

b.  x - 9 ln(x+9) + c

c.  ln(x+9) + c

d.  ln(x² +9) + c

e.  (1/2)ln(x² +9) + c

2.  Compute the indefinite integral  \( \int \frac{1}{x^2 +9}\) dx.

a.  (1/3) arctan(x/3) + c

b.  x - 9 ln(x+9) + c

c.  ln(x+9) + c

d.  ln(x² +9) + c

e.  (1/2)ln(x² +9) + c

3.  Compute the indefinite integral  \( \int \frac{x}{x^2 + 9} \) dx.

a.  (1/3) arctan(x/3) + c

b.  x - 9 ln(x+9) + c

c.  ln(x+9) + c

d.  ln(x² +9) + c

e.  (1/2)ln(x² +9) + c

4.  Compute the indefinite integral  \( \int \frac{x}{x+9} \)dx.

a.  (1/3) arctan(x/3) + c

b.  x - 9 ln(x+9) + c

c.  ln(x+9) + c

d.  ln(x² +9) + c

e.  (1/2)ln(x² +9) + c

5. Find the area enclosed between the curves \( y = 2^x\) and \( y = \frac{3}{2}x + 1\).

a.  \(\frac{2}{3} \)

b.  \( 5- \frac{3}{\ln 2}\)

c.  \( 1\)

d.  \( 2\)

e.  \( 4 - 5 \ln 2\)

6. Find the volume enclosed when you revolve the curve \( y = e^{\frac{x}{2}} \) around the \(x\)-axis between \(x = 0\) and \(x = 4\).

a.  \(\pi e^8 \)

b.  \(\pi (e^2 - 1)\)

c.  \(\pi (e^4 - 1)\)

d.  \(\pi e^4\)

e.  \( \pi\)

7. Find the volume of the solid obtained when you revolve the region enclosed between the curves \( y = x^2\) and \(y = 4\) around the \(x\)-axis.

a.  \(\frac{256\pi}{5} \)

b.  \(\frac{128\pi}{5}\)

c.  \(\pi (16 - \frac{1}{5})\)

d.  \(\pi (16 - \frac{1}{3})\)

e.  \( \pi (16 - \frac{1}{7})\)

8. Which integral would you have to evaluate to determine the length of the curve \(y = x^2\) between \(x = 0\), and \(x = 1\)?

a.  \(\pi\int_0^1 \sqrt{1 + x^2} dx\)

b.  \(\pi\int_0^1 \sqrt{1 + 4x^2} dx\)

c.  \(\pi\int_0^1 \sqrt{1 + 2x^2} dx\)

d.  \(\pi\int_0^1 \sqrt{1 + 2x} dx\)

e.  \(\pi\int_0^1 \sqrt{1 + x} dx\)

9. Find the surface area of the surface of rotation of the curve \(y = x\) between \(x = 0\) and \(x = 1\).

a.  \(\pi \)

b.  \(\sqrt{2}\pi\)

c.  \(2 \pi\)

d.  \(2\sqrt{2} \pi)\)

e.  \(4 \pi\)

10. Find the total work done if you push a rock from \(x = 0\) m to \(x = 4\) m if you're pushing with force \(F(x) = x\) N.

a.  \(2 J \)

b.  \(4 J\)

c.  \(8 J\)

d.  \(16 J\)

e.  \(32 J\)

For the next two problems, we consider a 2.00 m long metal bar which has density \(\rho(x) = 12.0 x^2 + 18.0\) kg/m between \(x = 0\), and \(x = 2\).

11. Find the total mass \( m\) of the metal bar.

a.  48 kg

b.  58 kg

c.  68 kg

d.  78 kg

e.  85 kg

12. Find the center of mass \(\overline{x}\) of the metal bar.

a.  1.00 m

b.  1.17 m

c.  1.20 m

d.  1.24 m

e.  1.35 m

13. \(\int \frac{2}{(x-1)^4} dx \)

a.  \(2\ln(x-1)^4 + c\)

b.  \(8\ln\mid x-1 \mid\)

c.  \(-\frac{2}{3(x-1)^3}+c\)

d.  \(-\frac{8}{(x-1)^5}+c\)

e.  \(8\ln(x-1)^4 + c\)

14. \(\int x^4 \ln x dx \)

a.  \(\frac{1}{5}x^5\ln x -\frac{1}{25}x^5+c\)

b.  \(\frac{1}{6}x^6\ln x - \frac{1}{6}x^6+c\)

c.  \(\frac{1}{6}x^6\ln x+c\)

d.  \(x^3+4x^3\ln x + c\)

e.  \(\frac{1}{6}x^6\ln x - \frac{1}{36}x^6+c\)