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Pustam | पुस्तम | পুস্তম🇳🇵<p>Integrals of inverse functions!</p><p>Proof without words (see image; credit: Jonathan Steinbuch, CC BY-SA 3.0, via Wikimedia Commons)...</p><p>For any montonic and invertible function \(f(x)\) in the interval \([a,b]\):<br>\[\displaystyle\int_a^bf(x)~ \mathrm dx+\int_{f(a)=c}^{f(b)=d}f^{-1}(x)~\mathrm dx=b\cdot f(b)-a\cdot f(a)=bd-ac\] </p><p>If \(F\) is an antiderivative of \(f\), then the antiderivatives of \(f^{-1}\) are:<br>\[\boxed{\displaystyle\int f^{-1}(y)~\mathrm dy=yf^{-1}(y)-F\circ f^{-1}(y)+C}\] <br>where \(C\) is an arbitrary constant (of integration), and \(\circ\) is the composition operator (function composition).</p><p>For example:<br>\[\begin{align*}\displaystyle\int \sin^{-1}(y) \, \mathrm dy &amp;= y\sin^{-1}(y) - (-\cos(\sin^{-1}(y)))+C\\ &amp;=y\sin^{-1}(y)+\sqrt{1-y^2}+C\end{align*}\]</p><p>\[\displaystyle\int \ln(y) \, dy = y\ln(y)-\exp(\ln(y)) + C= y\ln(y)-y + C.\]</p><p><a href="https://mathstodon.xyz/tags/Function" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Function</span></a> <a href="https://mathstodon.xyz/tags/InverseFunction" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>InverseFunction</span></a> <a href="https://mathstodon.xyz/tags/InverseFunctions" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>InverseFunctions</span></a> <a href="https://mathstodon.xyz/tags/Functions" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Functions</span></a> <a href="https://mathstodon.xyz/tags/Integral" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Integral</span></a> <a href="https://mathstodon.xyz/tags/Integrals" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Integrals</span></a> <a href="https://mathstodon.xyz/tags/Antiderivative" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Antiderivative</span></a> <a href="https://mathstodon.xyz/tags/Integration" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Integration</span></a> <a href="https://mathstodon.xyz/tags/Calculus" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Calculus</span></a> <a href="https://mathstodon.xyz/tags/FunctionComposition" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>FunctionComposition</span></a> <a href="https://mathstodon.xyz/tags/CompositeFunction" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>CompositeFunction</span></a>)</p>
Clément Boudereau🦀<p>To me <a href="https://fosstodon.org/tags/FunctionalProgramming" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>FunctionalProgramming</span></a> and <a href="https://fosstodon.org/tags/FunctionComposition" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>FunctionComposition</span></a> or <a href="https://fosstodon.org/tags/Type" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Type</span></a> DrivenDevelopment is like a <a href="https://fosstodon.org/tags/lego" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>lego</span></a> : each type created should conform the "system". The <a href="https://fosstodon.org/tags/LegoSystem" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>LegoSystem</span></a> are just simple rules so that it is reusable as much as possible and <a href="https://fosstodon.org/tags/DomainDrivenDesign" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>DomainDrivenDesign</span></a> is a good at creating those types.</p><p><a href="https://cboudereau.github.io/ContainerKata/#/2" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">cboudereau.github.io/Container</span><span class="invisible">Kata/#/2</span></a></p>