The function is odd
∫ − ∞ ∞ sin 2 x cos 4 x ln ∣ x ∣ tan 3 x d x = ?
The answer is 0.
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1 solution
This Integral is actually undefined because of the infinite number of vertical asymptotes that occur. This is like saying the integral from -infintity to infinity of 1/x is 0. But when split up the integral at x=0 to take into consideration the discontinuity, we see that I.t actually doesn’t even exist.
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In that case we work with the principal value. For instance, we could integrate over intervals [ − 2 π + ϵ , + 2 π − ϵ ] and take the limit ϵ → 0 .
An odd function satisfies the property f ( − x ) = − f ( x ) .
This applies to f ( x ) = sin 2 x cos 4 x ln ∣ x ∣ tan 3 x because f ( − x ) = sin 2 ( − x ) cos 4 ( − x ) ln ∣ − x ∣ tan 3 ( − x ) = ( − sin x ) 2 cos 4 x ln ∣ x ∣ ( − tan x ) 3 = sin 2 x cos 4 x ln ∣ x ∣ ⋅ − tan 3 x = − sin 2 x cos 4 x ln ∣ x ∣ tan 3 x = − f ( x ) .
WIth f odd, we also have ∫ − a a f ( x ) d x = ∫ − a 0 f ( x ) d x + ∫ 0 a f ( x ) d x = ∫ 0 a f ( − x ) d x + ∫ 0 a f ( x ) d x = − ∫ 0 a f ( x ) d x + ∫ 0 a f ( x ) d x = 0 . Let a → ∞ to conclude that the given integral is zero.