Which of the following statements are true? I. ∫ (f + g) dx = ∫ f dx + ∫ g dx. II. ∫ kf dx = k∫ f dx for a constant k. III. ∫ fg dx = ∫ f dx · ∫ g dx. IV. Two antiderivatives of the same function differ by a constant.

Question 9

MediumOriginal question

Which of the following statements are true?
I. $\int (f + g)\,dx = \int f\,dx + \int g\,dx$ .
II. $\int kf\,dx = k\int f\,dx$ for a constant $k$ .
III. $\int fg\,dx = \int f\,dx \cdot \int g\,dx$ .
IV. Two antiderivatives of the same function differ by a constant.

All four: I, II, III and IVCorrect answerYour answerI and II onlyCorrect answerYour answerI, II and IV onlyCorrect answerYour answerII, III and IV onlyCorrect answerYour answer

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CorrectNot quiteAnswer C

Correct answer: C

I and II are the two properties of 7.2.1, and IV holds because two functions with the same derivative differ by a constant.
III is false: with $f = g = x$ , $\int x^2\,dx = \frac{x^3}{3}$ , but $\frac{x^2}{2} \cdot \frac{x^2}{2} = \frac{x^4}{4}$ .

Concept: Properties of the indefinite integral
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsIntegration as the reverse of differentiation

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