# Inverse function integration formula

From Calculus

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## Contents

## Statement

### In terms of indefinite integrals

Suppose is a continuous one-one function. Then, we have:

where .

More explicitly, if is an antiderivative for , then:

This can be justified either directly or using integration by parts and integration by u-substitution.

### In terms of definite integrals

Suppose is a continuous one-one function on an interval. Suppose we are integrating on an interval of the form that lies in the range of . Then:

This simplifies to:

More explicitly, if is an antiderivative for , then:

## Significance

### Computational feasibility significance

Version type | Significance |
---|---|

indefinite integral | Given an antiderivative for a continuous one-one function , it is possible to explicitly write down an antiderivative for the inverse function in terms of and the antiderivative for . |

definite integral | Given an antiderivative for a continuous one-one function , and given knowledge of the values of at and , it is possible to explicitly compute the definite integral of on . |

## Examples

Original function | Domain on which it restricts to a one-one function | Inverse function for the restriction to that domain | Domain of inverse function (equals range of original function) | Antiderivative of original function | Antiderivative of inverse function | Explanation using inverse function integration formula | Alternate explanation using integration by parts |
---|---|---|---|---|---|---|---|

sine function | arc sine function | negative of cosine function, i.e., | We get . Now, use that is nonnegative on the range of and that to rewrite . | The usual method: <togledisplay>Take 1 as the part to integrate, get For the latter integration, put , get . The minus sign cancels with the outer minus sign, and we get the result.</toggledisplay> | |||

tangent function | arc tangent function | all real numbers | , same as | We get . Use that and simplify. | |||

exponential function | all real numbers | natural logarithm | exponential function | We get |