Trig symmetry.

Calculus Level 2

∫ − π π s i n ( x ) x 2 + 1 d x = \ \int _{ -\pi }^{ \pi }{ \frac { sin(x) }{ { x }^{ 2 }+1 } } dx =


The answer is 0.

Will McGlaughlin by Will McGlaughlin , 21, Australia

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1 solution

Will McGlaughlin
May 7, 2018

s i n ( − x ) ( − x ) 2 + 1 = − s i n ( x ) x 2 + 1 \large \frac { sin(-x) }{ { (-x })^{ 2 }+1 } =\frac { -sin(x) }{ { x }^{ 2 }+1 }

therefore it is an odd function, the integral is symmetrical and with evaluate to 0 \boxed{0}

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Can you explain how if a function is an odd function, it is symmetrical and how would it result that to 0?

Aman thegreat - 3 years, 1 month ago

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f(a) = -f(a), therefore each area above the x axis has a respective area under the x axis, which when evaluating the integral equals zero. If we were wanting to take the absolute area we would double the integral from zero to pi.

Will McGlaughlin - 3 years, 1 month ago

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Oh. Thanks

Aman thegreat - 3 years, 1 month ago

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