How can be helpful inverse functions determination by integral? <msup> f <mrow class=

kulisamilhh

kulisamilhh

Answered question

2022-05-03

How can be helpful inverse functions determination by integral?
f − 1 ( x ) = ∫ 1 f ′ ( f − 1 ( x ) ) d x + c .

Answer & Explanation

Eliza Flores

Eliza Flores

Beginner2022-05-04Added 16 answers

The formula will be useful if f ′ ( x ) can be expressed as a function of f ( x ). i.e there exists a function g ( y ) such that f ′ ( x ) = g ( f ( x ) ). Take the following scenario as an example f ( x ) = tan ⁡ ( x ), we know
d d x tan ⁡ ( x ) = 1 + tan ⁡ ( x ) 2   ⟺   g ( y ) = 1 + y 2 
Consider incorporating this into your calculation f ( 0 ) = 0, you get
f − 1 ( x ) = tan − 1 ⁡ ( x ) = ∫ 0 x d t f ′ ( f − 1 ( t ) ) = ∫ 0 x d t g ( t ) = ∫ 0 x d t 1 + t 2 
For | t | < 1, we have
1 1 + t 2 = ∑ n = 0 ∞ ( − 1 ) n t 2 n   ⟹   tan − 1 ⁡ ( x ) = ∑ n = 0 ∞ ( − 1 ) n ∫ 0 x t 2 n d x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 
Given the relationship, you can create a power series representation of tan − 1 ⁡ ( x ) for | x | < 1.

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