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Evaluating definite integrals worksheets Grade 12

Evaluate definite integrals with F(b) − F(a).

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How to do it

Find an antiderivative F, then work out F(top) − F(bottom).

∫−12(x2−x−4) dx=−212\int_{-1}^{2} (x^{2} - x - 4)\, dx = -\dfrac{21}{2}
F(x)=x33−x22−4xF(x) = \dfrac{x^{3}}{3} - \dfrac{x^{2}}{2} - 4xan antiderivative; no C is needed
(−223)−(196)=−212(-\dfrac{22}{3}) - (\dfrac{19}{6}) = -\dfrac{21}{2}F(2) − F(−1)

Example problems

  1. ∫14(3x3−4x2−x−1) dx=\int_{1}^{4} (3x^{3} - 4x^{2} - x - 1)\, dx =
  2. ∫−20(−4x3−5x2−5x+5) dx=\int_{-2}^{0} (-4x^{3} - 5x^{2} - 5x + 5)\, dx =
  3. ∫23(2x+5) dx=\int_{2}^{3} (2x + 5)\, dx =
  4. ∫02(x3−2x2−2x−4) dx=\int_{0}^{2} (x^{3} - 2x^{2} - 2x - 4)\, dx =

Integrals

  1. Antiderivatives with the power rule
  2. Integrals of 1/x, eˣ, sin and cos
  3. Integration by u-substitution
  4. Evaluating definite integrals
  5. Area under a curve
  6. Area between two curves
  7. Average value of a function