√(x)/√(a-x)の積分

√(x)/√(a-x)の積分

不定積分

$$\begin{array}{rcl}\int \frac{\sqrt{x}}{\sqrt{a-x}}\mathrm{d}x& =& \int x^{\frac{1}{2}}(a-x{)}^{-\frac{1}{2}}\mathrm{d}x\\ & =& \int u(a-u^2{)}^{-\frac{1}{2}}2u\mathrm{d}u\cdots u=\sqrt{x},\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{1}{2\sqrt{x}},\mathrm{d}x=2\sqrt{x}\mathrm{d}u=2u\mathrm{d}u\\ & =& 2\int u^2(a-u^2{)}^{-\frac{1}{2}}\mathrm{d}u\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& 2\int {\{\sqrt{a}\sin \left(\theta \right)\}}^2{[a-{\{\sqrt{a}\sin \left(\theta \right)\}}^2]}^{-\frac{1}{2}}\sqrt{a}\cos \left(\theta \right)\mathrm{d}\theta \cdots u=\sqrt{a}\sin \left(\theta \right),\frac{\mathrm{d}u}{\mathrm{d}\theta }=\sqrt{a}\cos \left(\theta \right),\mathrm{d}u=\sqrt{a}\cos \left(\theta \right)\mathrm{d}\theta \\ & =& 2\int a{\sin }^2\left(\theta \right){\{a-a{\sin }^2\left(\theta \right)\}}^{-\frac{1}{2}}\sqrt{a}\cos \left(\theta \right)\mathrm{d}\theta \\ & =& 2a\sqrt{a}\int {\sin }^2\left(\theta \right){\left[a\{1-{\sin }^2\left(\theta \right)\}\right]}^{-\frac{1}{2}}\cos \left(\theta \right)\mathrm{d}\theta \\ & =& 2a\sqrt{a}\int \sqrt{a}{\sin }^2\left(\theta \right)a^{-\frac{1}{2}}{\{1-{\sin }^2\left(\theta \right)\}}^{-\frac{1}{2}}\cos \left(\theta \right)\mathrm{d}\theta \\ & =& 2a\sqrt{a}{a}^{-\frac{1}{2}}\int {\sin }^2\left(\theta \right){\{1-{\sin }^2\left(\theta \right)\}}^{-\frac{1}{2}}\cos \left(\theta \right)\mathrm{d}\theta \\ & =& 2a\int {\sin }^2\left(\theta \right){\{1-{\sin }^2\left(\theta \right)\}}^{-\frac{1}{2}}\cos \left(\theta \right)\mathrm{d}\theta \\ & =& 2a\int {\sin }^2\left(\theta \right){\{{\cos }^2\left(\theta \right)\}}^{-\frac{1}{2}}\cos \left(\theta \right)\mathrm{d}\theta \\ & =& 2a\int {\sin }^2\left(\theta \right){\{\cos \left(\theta \right)\}}^{-1}\cos \left(\theta \right)\mathrm{d}\theta \\ & =& 2a\int {\sin }^2\left(\theta \right)\mathrm{d}\theta \\ & =& 2a\{\frac{1}{2}\theta -\frac{1}{2}\sin \left(\theta \right)\cos \left(\theta \right)\}\cdots \int {\sin }^2\left(\theta \right)\mathrm{d}\theta =\frac{1}{2}\{\theta -\sin \left(\theta \right)\cos \left(\theta \right)\}+C(C:\mathrm{積}\mathrm{分}\mathrm{定}\mathrm{数})\\ & =& 2a\{\frac{1}{2}{\sin }^{-1}\left(\frac{u}{\sqrt{a}}\right)-\frac{1}{2}\sin ({\sin }^{-1}\left(\frac{u}{\sqrt{a}}\right))\cos ({\sin }^{-1}\left(\frac{u}{\sqrt{a}}\right))\}\cdots u=\sqrt{a}\sin \left(\theta \right),\frac{u}{\sqrt{a}}=\sin \left(\theta \right),\theta ={\sin }^{-1}\left(\frac{u}{\sqrt{a}}\right)\\ & =& 2a\{\frac{1}{2}{\sin }^{-1}\left(\frac{u}{\sqrt{a}}\right)-\frac{1}{2}\left(\frac{u}{\sqrt{a}}\right)\sqrt{1-{\left(\frac{u}{\sqrt{a}}\right)}^2}\}\cdots \sin ({\sin }^{-1}\left(x\right))=x,\cos ({\sin }^{-1}\left(x\right))=\sqrt{1-x^2}\\ & =& 2a\frac{1}{2}\{{\sin }^{-1}\left(\frac{u}{\sqrt{a}}\right)-\left(\frac{u}{\sqrt{a}}\right)\sqrt{1-{\left(\frac{u}{\sqrt{a}}\right)}^2}\}\\ & =& a\{{\sin }^{-1}\left(\frac{u}{\sqrt{a}}\right)-\left(\frac{u}{\sqrt{a}}\right)\sqrt{1-{\left(\frac{u}{\sqrt{a}}\right)}^2}\}\\ & =& a\{{\sin }^{-1}\left(\frac{\sqrt{x}}{\sqrt{a}}\right)-\left(\frac{\sqrt{x}}{\sqrt{a}}\right)\sqrt{1-{\left(\frac{\sqrt{x}}{\sqrt{a}}\right)}^2}\}\cdots u=\sqrt{x}\\ & =& a\{{\sin }^{-1}\left(\sqrt{\frac{x}{a}}\right)-\frac{\sqrt{x}}{\sqrt{a}}\sqrt{1-\frac{x}{a}}\}\\ & =& a\{{\sin }^{-1}\left(\sqrt{\frac{x}{a}}\right)-\frac{\sqrt{x}}{\sqrt{a}}\sqrt{\frac{a-x}{a}}\}\\ & =& a\{{\sin }^{-1}\left(\sqrt{\frac{x}{a}}\right)-\frac{\sqrt{x}}{\sqrt{a}}\frac{\sqrt{a-x}}{\sqrt{a}}\}\cdots \sqrt{\frac{A}{B}}={\left(\frac{A}{B}\right)}^{\frac{1}{2}}={\left(A\frac{1}{B}\right)}^{\frac{1}{2}}=A^{\frac{1}{2}}{\left(\frac{1}{B}\right)}^{\frac{1}{2}}=A^{\frac{1}{2}}{\left(B^{-1}\right)}^{\frac{1}{2}}=A^{\frac{1}{2}}{B}^{-\frac{1}{2}}=\sqrt{A}\frac{1}{\sqrt{B}}=\frac{\sqrt{A}}{\sqrt{B}}\\ & =& a\{{\sin }^{-1}\left(\sqrt{\frac{x}{a}}\right)-\frac{\sqrt{x}\sqrt{a-x}}{a}\}\\ & =& a{\sin }^{-1}\left(\sqrt{\frac{x}{a}}\right)-a\frac{\sqrt{x}\sqrt{a-x}}{a}\\ & =& a{\sin }^{-1}\left(\sqrt{\frac{x}{a}}\right)-\sqrt{x}\sqrt{a-x}+C\cdots C:\mathrm{積}\mathrm{分}\mathrm{定}\mathrm{数}\\ & =& a{\sin }^{-1}\left({\left(\frac{x}{a}\right)}^{\frac{1}{2}}\right)-x^{\frac{1}{2}}{(a-x)}^{\frac{1}{2}}+C\end{array}$$

定積分

$$\begin{array}{rcl}{\int }_0^a\frac{\sqrt{x}}{\sqrt{a-x}}\mathrm{d}x& =& {[a{\sin }^{-1}\left({\left(\frac{x}{a}\right)}^{\frac{1}{2}}\right)-x^{\frac{1}{2}}{(a-x)}^{\frac{1}{2}}]}_0^a\\ & =& (a{\sin }^{-1}\left({\left(\frac{a}{a}\right)}^{\frac{1}{2}}\right)-a^{\frac{1}{2}}{(a-a)}^{\frac{1}{2}})-(a{\sin }^{-1}\left({\left(\frac{0}{a}\right)}^{\frac{1}{2}}\right)-0^{\frac{1}{2}}{(a-0)}^{\frac{1}{2}})\\ & =& (a{\sin }^{-1}\left(1\right)-0)-(0-0)\cdots {\sin }^{-1}\left(0\right)=0,0^A=0\\ & =& a\frac{\pi }{2}\cdots {\sin }^{-1}\left(1\right)=\frac{\pi }{2}\\ & =& \frac{a\pi }{2}\end{array}$$