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Integration Technique Review

  1. \(\int_1^2 (9x^2 - 4x + 1) \ln x \, dx\)

    \[ \begin{align} &= \int_1^2 9x^2 \ln x \, dx - \int_1^2 4x \ln x \, dx + \int_1^2 \ln x \, dx \\ &= \left[ 3x^3 \ln x - \int 3x^2 \, dx \right]_1^2 - \left[ 2x^2 \ln x - \int 2x \, dx \right]_1^2 + \left[ x \ln x - x \right]_1^2 \\ &= \left[ 3x^3 \ln x - x^3 \right]_1^2 - \left[ 2x^2 \ln x - x^2 \right]_1^2 + \left[ x \ln x - x \right]_1^2 \\ &= \left[ 24 \ln 2 - 8 - 3 \ln 1 + 1 \right] - \left[ 8 \ln 2 - 4 - 4 \ln 1 + 1 \right] + \left[ 2 \ln 2 - 2 - \ln 1 + 1 \right] \\ &= 24 \ln 2 - 8 - 3 + 1 - 8 \ln 2 + 4 + 4 - 1 + 2 \ln 2 - 2 - 1 + 1 \\ &= 24 \ln 2 - 8 \ln 2 + 2 \ln 2 - 8 + 4 + 4 - 3 - 1 - 2 - 1 + 1 \\ &= 18 \ln 2 - 7 \quad \blacksquare \end{align} \]

  2. The function \(f\) has a continuous derivative. The table below gives values of \(f\) and its derivative for \(x=0\) and \(x=4\). If \(\int_0^4 f(x) \, dx = 8\), what is the value of \(\int_0^4 xf'(x) \, dx\)?

    \(x\) \(f(x)\) \(f'(x)\)
    \(0\) \(2\) \(5\)
    \(4\) \(-3\) \(11\)
    \[ \begin{align} u = x \quad & dv = f'(x) \, dx \\ du = dx \quad & v = f(x) \end{align} \]
    \[ \begin{align} \int_0^4 xf'(x) \, dx &= \left[ x f(x) \right]_0^4 - \int_0^4 f(x) \, dx \\ &= 4f(4) - 0f(0) - 8 \\ &= 4(-3) - 0(2) - 8 \\ &= -12 - 8 \\ &= -20 \quad \blacksquare \end{align} \]

  3. \(\int x \sin(6x) \, dx\)

    \[ \begin{align} u = x \quad & dv = \sin(6x) \, dx \\ du = dx \quad & v = -\frac{1}{6} \cos(6x) \end{align} \]
    \[ \begin{align} \int x \sin(6x) \, dx &= -x \frac{1}{6} \cos(6x) - \int -\frac{1}{6} \cos(6x) \, dx \\ &= -\frac{x}{6} \cos(6x) + \frac{1}{6} \int \cos(6x) \, dx \\ &= -\frac{x}{6} \cos(6x) + \frac{1}{6} \left[ \frac{1}{6} \sin(6x) \right] \\ &= -\frac{x}{6} \cos(6x) + \frac{1}{36} \sin(6x) + C \quad \blacksquare \end{align} \]

  4. \(x f(x) \, dx\)

    \[ \begin{align} u = x \quad & dv = f(x) \, dx \\ \end{align} \]

  5. If \(\int f(x) \sin x \, dx = - f(x) \cos x + \int 3x^2 \cos x \, dx\), then \(f(x)\) could be

  6. \(\int_1^e x^4 \ln x \, dx\)
  7. \(\int \frac{7x}{(2x-3)(x+2)} \, dx\)
  8. \(\int \frac{1}{x^2 - 7x + 10} \, dx\)
  9. Which of the following expressions is equal to \(\int_0^2 \frac{17x+4}{3x^2-7x-6} \, dx\)?
  10. \(\int \frac{2x}{(x+2)(x+1)} \, dx\)
  11. Let \(R\) be the region between the graph of \(y=e^{-2x}\) and the \(x\)-axis for \(x \geq 3\). The area of \(R\) is
  12. \(\int_0^3 \frac{dx}{(1-x)^2}\)
  13. \(\int_1^{\infty} xe^{-x^2} \, dx\)
  14. What are all values of \(p\) for which \(\int_1^{\infty} \frac{1}{x^{3p + 1}} \, dx\) converges?
  15. An antiderivative of \(\frac{e^x}{e^x - 1}\) is \(\ln |e^x - 1|\). Which of the following statements about the integral \(\int_{-2}^2 \frac{e^x}{e^x - 1} \, dx\) is true?
  16. \(\int_0^4 \frac{1}{\sqrt{x}(1+\sqrt{x})} \, dx\)