Primitive function/Inverse function/Section

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Let denote a bijective differentiable function, and let denote a primitive function for . Then

is a primitive function for the inverse function .

Differentiating, using fact and fact, yields

The graph with its inverse function, and the areas relevant for the computation of the integral of the inverse function.

For this statement, there exists also an easy geometric explanation. When is a strictly increasing continuous function (and therefore induces a bijection between and ), then the following relation between the areas holds:

or, equivalently,

For the primitive function of with starting point , we have, if denotes a primitive function for , the relation

where is a constant of integration.

We compute a primitive function for , using fact. A primitive function of tangent is


is a primitive function for .