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>\[\begin{align*}\displaystyle\int \sin^{-1}(y) \, \mathrm dy &= y\sin^{-1}(y) - (-\cos(\sin^{-1}(y)))+C\\ &=y\sin^{-1}(y)+\sqrt{1-y^2}+C\end{align*}\]</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>