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Integration

4.1 Antiderivatives and Indefinite Integration

Focus on antiderivatives and the integration constant, reverse the power rule, trigonometric antiderivatives.

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Ideas in this lesson

01Antiderivatives and the integration constantWorked steps · five questions02Reverse the power ruleWorked steps · five questions03Trigonometric antiderivativesWorked steps · five questions
Idea 1 of 3

Antiderivatives and the integration constant

An indefinite integral describes a family of functions differing by a constant. Differentiate to check an antiderivative.

Worked example 1

Let F′(x)=2xF'(x) = 2x and F(0)=1F(0) = 1. Find F(1)F(1).
  1. F(x)=22x2+CF(x) = \dfrac{2}{2}x^{2} + C
  2. F(0)=C=1F(0) = C = 1
  3. F(1)=22+1=2F(1) = \dfrac{2}{2} + 1 = 2

Answer: 22

Worked example 2

Let F′(x)=3x2F'(x) = 3x^{2} and F(0)=2F(0) = 2. Find F(1)F(1).
  1. F(x)=33x3+CF(x) = \dfrac{3}{3}x^{3} + C
  2. F(0)=C=2F(0) = C = 2
  3. F(1)=33+2=3F(1) = \dfrac{3}{3} + 2 = 3

Answer: 33

Idea 2 of 3

Reverse the power rule

Raise the power and divide by the new power when the exponent is not −1-1. The exponent −1-1 uses a logarithm.

Worked example 1

Evaluate ∫02x dx\int_{0}^{2} x\,dx.
  1. F(x)=x22F(x) = \dfrac{x^{2}}{2}
  2. F(2)−F(0)=222−0=2F(2) - F(0) = \dfrac{2^{2}}{2} - 0 = 2

Answer: 22

Worked example 2

Evaluate ∫03x2 dx\int_{0}^{3} x^{2}\,dx.
  1. F(x)=x33F(x) = \dfrac{x^{3}}{3}
  2. F(3)−F(0)=333−0=9F(3) - F(0) = \dfrac{3^{3}}{3} - 0 = 9

Answer: 99

Idea 3 of 3

Trigonometric antiderivatives

Remember the signs: an antiderivative of sin⁡x\sin x is −cos⁡x-\cos x, and an antiderivative of cos⁡x\cos x is sin⁡x\sin x.

Worked example 1

Evaluate ∫0π1sin⁡x dx\int_{0}^{\pi } 1\sin x\,dx.
  1. F(x)=−1cos⁡xF(x) = -1 \cos x
  2. F(π)−F(0)=1−(−1)=2F(\pi ) - F(0) = 1 - (-1) = 2

Answer: 22

Worked example 2

Evaluate ∫0π22cos⁡x dx\int_{0}^{\dfrac{\pi }{2}} 2\cos x\,dx.
  1. F(x)=2sin⁡xF(x) = 2 \sin x
  2. F(π2)−F(0)=2−0=2F(\dfrac{\pi }{2}) - F(0) = 2 - 0 = 2

Answer: 22

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §4.1,p4.1, p. 244244. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.