Which of the following statements are true? I. ∫ e^2x dx = e^2x/2 + C. II. ∫ 1/x^2 dx = log x^2 + C. III. ∫ k f(x) dx = k∫ f(x) dx for a constant k. IV. ∫ f(x)g(x) dx = ∫ f(x) dx · ∫ g(x) dx.

Question 15

MediumOriginal question

Which of the following statements are true?
I. $\int e^{2x}\,dx = \frac{e^{2x}}{2} + C$ .
II. $\int \frac{1}{x^{2}}\,dx = \log x^{2} + C$ .
III. $\int k f(x)\,dx = k\int f(x)\,dx$ for a constant $k$ .
IV. $\int f(x)g(x)\,dx = \int f(x)\,dx \cdot \int g(x)\,dx$ .

I and III onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answerI onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answer

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CorrectNot quiteAnswer A

Correct answer: A

I: $\frac{d}{dx}\frac{e^{2x}}{2} = e^{2x}$ . III: a constant factor comes out of an integral.
II is false: $\int x^{-2}\,dx = -\frac{1}{x} + C$ . IV is false: there is no product rule for integrals (try $f = g = x$ ).

Worth noting

$\log$ appears only for the power $-1$ ; every other power, $-2$ included, follows $\frac{x^{n + 1}}{n + 1}$ .

Concept: Properties of the indefinite integral
Typical timeTypical: 120 s
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The rule this testsIntegration as the reverse of differentiation

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