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Inverse Function Integration

Inverse function integration is an indefinite integration technique. While simple, it is an interesting application of integration by parts.

If f and f^(-1) are inverses of each other on some closed interval, then

intf(x)dx=xf(x)-intf^(-1)(f(x))f^'(x)dx,
(1)

so

intf(x)dx=xf(x)-G(f(x)),
(2)

where

G(x)=intf^(-1)(x)dx.
(3)

Therefore, if it is possible to find an inverse f^(-1) of f, integrate f^(-1), make the replacement x->f(x), and subtract the result from xf(x) to obtain the result for the original integral intf(x)dx.

If f and f^(-1) are elementary on some closed interval, then intf(x)dx is elementary iff intf^(-1)(x)dx is elementary.

See also

Integration, Integration by Parts

This entry contributed by Bhuvanesh Bhatt

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References

Key, E. "Disks, Shells, and Integrals of Inverse Functions." Coll. Math. J. 25, 136-138, 1994.Parker, F. D. "Integrals of Inverse Functions." Amer. Math. Monthly 62, 439-440, 1955.

Referenced on Wolfram|Alpha

Inverse Function Integration

Cite this as:

Bhatt, Bhuvanesh. "Inverse Function Integration." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InverseFunctionIntegration.html

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