x/√(a-x)の積分

x/√(a-x)の積分

不定積分

$$\begin{array}{rcl}\int \frac{x}{\sqrt{a-x}}\mathrm{d}x& =& \int x(a-x{)}^{-\frac{1}{2}}\mathrm{d}x\\ & =& \int x{\{-2(a-x{)}^{\frac{1}{2}}\}}^{\prime }\mathrm{d}x\cdots \int (a-x{)}^{-\frac{1}{2}}\mathrm{d}x=-2(a-x{)}^{\frac{1}{2}}+C\\ & =& -2x(a-x{)}^{\frac{1}{2}}-\int -2(a-x{)}^{\frac{1}{2}}\mathrm{d}x\cdots \int f^{\prime }(x)g(x)\mathrm{d}x=fg-\int f{g}^{\prime }\mathrm{d}x\\ & =& -2x(a-x{)}^{\frac{1}{2}}+2\int (a-x{)}^{\frac{1}{2}}\mathrm{d}x\\ & =& -2x(a-x{)}^{\frac{1}{2}}+2\{\frac{1}{\frac{1}{2}+1}(a-x{)}^{\frac{1}{2}+1}(-1)\}\\ & =& -2x(a-x{)}^{\frac{1}{2}}+2\{\frac{-1}{\frac{3}{2}}(a-x{)}^{\frac{3}{2}}\}\\ & =& -2x(a-x{)}^{\frac{1}{2}}+2\{\frac{-2}{3}(a-x{)}^{\frac{3}{2}}\}\\ & =& -2x(a-x{)}^{\frac{1}{2}}-\frac{4}{3}(a-x{)}^{\frac{3}{2}}\\ & =& -2x(a-x{)}^{\frac{1}{2}}-\frac{4}{3}(a-x)(a-x{)}^{\frac{1}{2}}\\ & =& (a-x{)}^{\frac{1}{2}}\{-2x-\frac{4}{3}(a-x)\}\\ & =& (a-x{)}^{\frac{1}{2}}(-2x-\frac{4}{3}a+\frac{4}{3}x)\\ & =& (a-x{)}^{\frac{1}{2}}(-\frac{6}{3}x-\frac{4}{3}a+\frac{4}{3}x)\\ & =& (a-x{)}^{\frac{1}{2}}(-\frac{2}{3}x-\frac{4}{3}a)\\ & =& -\frac{2}{3}(a-x{)}^{\frac{1}{2}}(x+2a)+C\cdots C:\mathrm{積}\mathrm{分}\mathrm{定}\mathrm{数}\\ & =& -\frac{2}{3}\sqrt{a-x}(x+2a)+C\end{array}$$

定積分

$$\begin{array}{rcl}{\int }_0^a\frac{x}{\sqrt{a-x}}\mathrm{d}x& =& {[-\frac{2}{3}\sqrt{a-x}(x+2a)]}_0^a\\ & =& (-\frac{2}{3}\sqrt{a-a}(a+2a))-(-\frac{2}{3}\sqrt{a-0}(0+2a))\\ & =& 0-(-\frac{2}{3}\sqrt{a}\left(2a\right))\\ & =& 0-(-\frac{4}{3}{\sqrt{a}}^3)\\ & =& \frac{4}{3}{a}^{\frac{3}{2}}\end{array}$$