Which of the following statements are true? I. ∫₋ₐ^a f(x) dx = 0 whenever f is odd. II. ∫₋ₐ^a f(x) dx = 2∫₀^a f(x) dx whenever f is even. III. ∫ₐ^b f(x) dx = ∫ₐ^b f(a + b - x) dx. IV. ∫ₐ^b f(x)g(x) dx = ∫ₐ^b f(x) dx · ∫ₐ^b g(x) dx.

Question 42

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Which of the following statements are true?
I. $\int_{-a}^{a} f(x)\,dx = 0$ whenever $f$ is odd.
II. $\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx$ whenever $f$ is even.
III. $\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$ .
IV. $\int_a^b f(x)g(x)\,dx = \int_a^b f(x)\,dx \cdot \int_a^b g(x)\,dx$ .

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

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CorrectNot quiteAnswer D

Correct answer: D

I, II and III are properties of definite integrals: the odd and even rules, and the reflection $x \to a + b - x$ , which only reverses the direction of travel across $[a, b]$ .
IV is false: on $[0, 1]$ with $f = g = x$ , $\int_0^1 x^2\,dx = \frac{1}{3}$ , but $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ .

Concept: Properties of definite integrals
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The rule this testsProperties of definite integrals

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