Notes & theory for this topic

Integration

Write your working on paper. Reveal each answer only after you've had a go.

1. Integrate the following expressions (give the indefinite integral, and don't forget + C).
  1. \(I = \int \left(8x^{4} - 8x^{3} + 5x^{2} + 4x - 6\right)\,dx\)
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    Reverse the power rule: add one to each exponent and divide by the new exponent.
    \(I = \frac{8}{5}x^{5} - 2x^{4} + \frac{5}{3}x^{3} + 2x^{2} - 6x + C\)
  2. \(I = \int 8\sin x\,dx\)
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    The integral of sin is −cos.
    \(I = -8\cos x + C\)
  3. \(I = \int 5\cos x\,dx\)
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    The integral of cos is sin.
    \(I = 5\sin x + C\)
  4. \(I = \int -e^{x}\,dx\)
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    e raised to x integrates to itself.
    \(I = -e^{x} + C\)
2. Integrate using the reverse chain rule.
  1. \(I = \int -3\sin\left(3x - 5\right)\,dx\)
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    Let \(u = 3x - 5\), so \(du/dx = 3\); divide by 3 when you integrate.
    \(I = \cos\left(3x - 5\right) + C\)
  2. \(I = \int -\cos\left(3x + 1\right)\,dx\)
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    Let \(u = 3x + 1\), so \(du/dx = 3\); divide by 3 when you integrate.
    \(I = -\frac{1}{3}\sin\left(3x + 1\right) + C\)
  3. \(I = \int 2e^{4x + 7}\,dx\)
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    Let \(u = 4x + 7\), so \(du/dx = 4\); divide by 4 when you integrate.
    \(I = \frac{1}{2}e^{4x + 7} + C\)
  4. \(I = \int \frac{-7}{5x - 6}\,dx\)
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    Let \(u = 5x - 6\), so \(du/dx = 5\); divide by 5 when you integrate.
    \(I = -\frac{7}{5}\ln\left|5x - 6\right| + C\)
3. Evaluate the following definite integrals.
  1. \(I = \int_{1}^{2} \left(5x^{2} - 8x + 7\right)\,dx\)
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    Integrate to \(\frac{5}{3}x^{3} - 4x^{2} + 7x\), then evaluate at x=2 minus at x=1.
    \(I = \frac{20}{3} \approx 6.667\)
  2. \(I = \int_{0}^{0.5} \sin\left(3x - 1\right)\,dx\)
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    Antiderivative \(-\frac{1}{3}\cos\left(3x - 1\right)\); evaluate between the limits.
    \(I = -0.1124\)
  3. \(I = \int_{0}^{0.5} \cos\left(x - 1\right)\,dx\)
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    Antiderivative \(\frac{1}{1}\sin\left(x - 1\right)\); evaluate between the limits.
    \(I = 0.3620\)
  4. \(I = \int_{0}^{1} e^{x - 1}\,dx\)
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    Antiderivative \(\frac{1}{1}e^{x - 1}\); evaluate between the limits.
    \(I = 0.6321\)
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