Be careful! Is the function always continuous?

Calculus Level 3

Which of the following statements is/are true?

(1): ∫ 0 2 π sin ⁡ x d x = 0 \displaystyle \int_0^{2\pi} \sin x \, dx = 0 .
(2): ∫ 0 2 π cos ⁡ x d x = 0 \displaystyle \int_0^{2\pi} \cos x \, dx = 0 .
(3): ∫ 0 2 π tan ⁡ x d x = 0 \displaystyle \int_0^{2\pi} \tan x \, dx = 0 .

Statements 1 and 2 only Statements 2 and 3 only Statements 1 and 3 only All 3 statements are true

Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Tom Engelsman
Feb 27, 2017

The function f ( x ) = t a n ( x ) f(x) = tan(x) is discontinuous at x = π 2 , 3 π 2 x = \frac{\pi}{2}, \frac{3\pi}{2} , and the #3 definite integral cannot be evaluated as is. Hence, only integrals #1 and #2 are correct.

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