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October 15

Indefinite integral of inverse function

If we know the indefinite integral g(x) = $\int f(x)dx,$ can we find the indefinite integral $\int f^{-1}(y)dy$ directly from g(x) without performing an integration? Duoduoduo (talk) 23:20, 15 October 2012 (UTC)[reply]

Integration by parts shows that

\int _{f(a)}^{f(x)}f^{-1}(t)dt+\int _{a}^{x}f(s)ds=xf(x)-af(a).

From this you should be able to figure out whatever you need to.Sławomir Biały (talk) 00:10, 16 October 2012 (UTC)[reply]

So

\int f^{-1}(x)dx=xf^{-1}(x)-g(f(x))+C

; example:

\int \arcsin {x}\,dx=x\arcsin {x}+\cos {(\arcsin {x})}+C=x\arcsin {x}+{\sqrt {1-\sin ^{2}{(\arcsin {x})}}}+C=x\arcsin {x}+{\sqrt {1-x^{2}}}+C

Ssscienccce (talk) 06:33, 17 October 2012 (UTC)[reply]

Thanks. Am I correct in assuming there's a typo in your first equation, Ssscienccce -- f(x) should be f^-1(x)? Duoduoduo (talk) 20:56, 17 October 2012 (UTC)[reply]