Brown University Mathematics Department
Self-Graded Calculus Placement Exam

SECTIONS B AND C

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SECTION B

Question #9: Find \(\displaystyle \int_0^1 (x^7-x^3) \, dx\).

  1. \(\displaystyle\frac{1}{8}\)
  2. \(\displaystyle-\frac{1}{8}\)
  3. \(\displaystyle-\frac{1}{4}\)
  4. 4
  5. None of the above answers

Question #10: Suppose a continuous function \(F(x)\) has derivative \(F'(x)=e^{3x}-\sin x\), and \(\displaystyle F(0)=\frac{1}{3}\). Find the function \(F(x)\).

  1. \(\displaystyle 3e^{3x} - \cos x + \frac{1}{3}\)
  2. \(\displaystyle 3e^{3x} - \cos x - \frac{5}{3}\)
  3. \(\displaystyle\frac{e^{3x}}{3} + \cos x + 1\)
  4. \(\displaystyle\frac{e^{3x}}{3} + \cos x - 1\)
  5. None of the above answers

Question #11: Which of the following quantities is equal to \(\displaystyle \int^3_2 {x \over {(x^2 - 1)^2}} dx\)?

  1. \(\displaystyle \left(-{1 \over 6}\right) \left({1 \over 8^3} - {1 \over 3^3}\right)\)
  2. \(\displaystyle {1 \over 2}\left(\ln 8 - \ln 3\right)\)
  3. \(\displaystyle 2\left({1 \over 8} - {1 \over 3}\right)\)
  4. \(\displaystyle \left(-{1 \over 2}\right) \left({1 \over 8} - {1 \over 3}\right)\)
  5. None of the above answers

Question #12: Use the substitution \(u = {\sqrt{x}}\) to change \(\displaystyle \int {{\sqrt {x} }\over {{\sqrt{\sqrt{x} - 1}}}} dx\) into an integral using the variable \(u\).

  1. \(\displaystyle \int {{2u^2 du} \over {{\sqrt{u - 1}}}}\)
  2. \(\displaystyle \int {{1 \over 2} {\sqrt{u}} \over {\sqrt{u - 1}}} dx\)
  3. \(\displaystyle \int {{1 \over 2} {\sqrt{u}} \over {\sqrt{u - 1}}} dx\)
  4. \(\displaystyle \int {{u} \over {\sqrt{u - 1}}} dx\)
  5. None of the above answers

SECTION C

Question #13: Find \(\displaystyle \int x e^{-x} dx\).

  1. \(\displaystyle {-x^2 \over 2} e^{-x} + C\)
  2. \(e^{{-x^2/2}} + C\)
  3. \(-xe^{-x} - e^{-x} + C\)
  4. \(-xe^{-x} + e^{-x} + C\)
  5. None of the above answers

Question #14: Find \(\displaystyle \int^\infty_1 {1 \over {x^3}}\).

  1. integral does not exist
  2. \(\displaystyle {1 \over 2}\)
  3. \(\displaystyle {1 \over 4}\)
  4. 1
  5. None of the above answers

Question #15: Find \(\displaystyle \int^{\pi/2}_0 \cos^5 x \sin x dx\).

  1. \(\displaystyle {1 \over 6}\)
  2. \(\displaystyle -{1 \over 6}\)
  3. \(\displaystyle {1 \over 5}\)
  4. 0
  5. None of the above answers

Question #16: Find \(\displaystyle \int^1_0 {1 \over {x^2 - 4}} dx\). Hint: Use the method of partial fractions to write \(\displaystyle {1 \over {x^2 - 4}} = {A \over {x - a}} + {B \over {x - b}}\) for certain constants \(A, B, a, b\). (\(\ln\) denotes the natural logarithm.)

  1. \(\displaystyle -{1 \over 2} \ln 3\)
  2. \(\displaystyle -{1 \over 4} \ln 3\)
  3. \(\displaystyle {1 \over 2} \ln 3 - \ln 2\)
  4. \(\displaystyle {1 \over 2} \ln 3 + \ln 2\)
  5. None of the above answers