Let f ( x ) = 2 x + sin ( 2 π x ) and g be the inverse function of f .
Compute ∫ 3 4 g ( x ) d x .
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∫ 3 4 g ( x ) d x = ∫ g ( 3 ) g ( 4 ) x d f ( x ) = ∫ 1 2 x ( 2 + 2 π cos ( 2 π x ) ) d x = ∫ 1 2 2 x d x + ∫ 1 2 2 π x cos ( 2 π x ) d x = x 2 + x sin ( 2 π x ) − ∫ sin ( 2 π x ) d x ∣ ∣ ∣ ∣ 1 2 = x 2 + x sin ( 2 π x ) + π 2 cos ( 2 π x ) ∣ ∣ ∣ ∣ 1 2 = 4 − 1 + 0 − 1 − π 2 − 0 = 2 − π 2 ≈ 1 . 3 6 3 Note that f ( x ) = 2 x + sin ( 2 π x ) ⟹ d f ( x ) = ( 2 + 2 π cos ( 2 π x ) ) d x By integration by parts
Doing a numerical integration over a meta-function which computes the inverse of a function numerically gives 1.36338022763213.
Thank you for the identity. See Integral of inverse functions .
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Using the beautiful property that
∫ a b f ( x ) d x + ∫ f ( a ) f ( b ) f − 1 ( x ) d x = b f ( b ) − a f ( a )
We have,
∫ 3 4 g ( x ) d x = 5 − ∫ 1 2 ( 2 x + sin ( 2 π x ) ) d x
which simply evaluates to 1 . 3 6 3 . . .