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. \(I = \int \left(2x^{4} - 5x^{3} + 5x^{2} - 8x + 8\right)\,dx\)
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    Reverse the power rule: add one to each exponent and divide by the new exponent.
    \(I = \frac{2}{5}x^{5} - \frac{5}{4}x^{4} + \frac{5}{3}x^{3} - 4x^{2} + 8x + C\)
  2. \(I = \int 6\sin x\,dx\)
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    The integral of sin is −cos.
    \(I = -6\cos x + C\)
  3. \(I = \int -8\cos x\,dx\)
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    The integral of cos is sin.
    \(I = -8\sin x + C\)
  4. \(I = \int -6e^{x}\,dx\)
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    e raised to x integrates to itself.
    \(I = -6e^{x} + C\)
2. Integrate using the reverse chain rule.
  1. \(I = \int -5\sin\left(7x + 2\right)\,dx\)
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    Let \(u = 7x + 2\), so \(du/dx = 7\); divide by 7 when you integrate.
    \(I = \frac{5}{7}\cos\left(7x + 2\right) + C\)
  2. \(I = \int 7\cos\left(2x - 1\right)\,dx\)
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    Let \(u = 2x - 1\), so \(du/dx = 2\); divide by 2 when you integrate.
    \(I = \frac{7}{2}\sin\left(2x - 1\right) + C\)
  3. \(I = \int -3e^{7x + 3}\,dx\)
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    Let \(u = 7x + 3\), so \(du/dx = 7\); divide by 7 when you integrate.
    \(I = -\frac{3}{7}e^{7x + 3} + C\)
  4. \(I = \int \frac{1}{4x + 1}\,dx\)
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    Let \(u = 4x + 1\), so \(du/dx = 4\); divide by 4 when you integrate.
    \(I = \frac{1}{4}\ln\left|4x + 1\right| + C\)
3. Evaluate the following definite integrals.
  1. \(I = \int_{-2}^{-1} \left(x^{2} - 2x + 8\right)\,dx\)
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    Integrate to \(\frac{1}{3}x^{3} - x^{2} + 8x\), then evaluate at x=-1 minus at x=-2.
    \(I = \frac{40}{3} \approx 13.33\)
  2. \(I = \int_{0}^{0.5} \sin\left(2x - 1\right)\,dx\)
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    Antiderivative \(-\frac{1}{2}\cos\left(2x - 1\right)\); evaluate between the limits.
    \(I = -0.2298\)
  3. \(I = \int_{0}^{1.5} \cos\left(2x\right)\,dx\)
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    Antiderivative \(\frac{1}{2}\sin\left(2x\right)\); evaluate between the limits.
    \(I = 0.07056\)
  4. \(I = \int_{0}^{1} e^{x}\,dx\)
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    Antiderivative \(\frac{1}{1}e^{x}\); evaluate between the limits.
    \(I = 1.718\)
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