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Worked solution

Practise exact antiderivatives

Without the use of a calculator, evaluate (a) ∫025(2x−1)4 dx\int_0^2 5(2x-1)^4\,dx, (b) ∫−1013x+14 dx\int_{-1}^0\frac1{\sqrt{3x+14}}\,dx.

CJC · Lecture · Self-Practice 1

(a)

An antiderivative is 12(2x−1)5\frac12(2x-1)^5. Hence [12(2x−1)5]02=12(35−(−1)5)=122[\frac12(2x-1)^5]_0^2=\frac12(3^5-(-1)^5)=122.

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(b)

An antiderivative is 233x+14\frac23\sqrt{3x+14}. Subtract the lower endpoint: [233x+14]−10=23(14−11)[\frac23\sqrt{3x+14}]_{-1}^0=\frac23(\sqrt{14}-\sqrt{11}).

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Common mistakes

Account for the inner derivative: the factor 2 or 3 changes the antiderivative coefficient.

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