x/√(a-x)の積分

x/√(a-x)の積分

不定積分

$$ \begin{eqnarray} \int \frac{x}{\sqrt{a-x}}\mathrm{d}x &=&\int x(a-x)^{-\frac{1}{2}}\mathrm{d}x \\&=&\int x\left\{-2(a-x)^{\frac{1}{2}}\right\}^\prime \mathrm{d}x \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/1a-x.html}{\int (a-x)^{-\frac{1}{2}} \mathrm{d}x=-2(a-x)^{\frac{1}{2}}+C} \\&=&-2x(a-x)^{\frac{1}{2}}-\int -2(a-x)^{\frac{1}{2}} \mathrm{d}x \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/02/blog-post_7.html}{\int f^\prime(x)g(x) \mathrm{d}x= fg-\int fg^\prime \mathrm{d}x} \\&=&-2x(a-x)^{\frac{1}{2}}+2\int (a-x)^{\frac{1}{2}} \mathrm{d}x \\&=&-2x(a-x)^{\frac{1}{2}}+2\left\{\frac{1}{\frac{1}{2}+1}(a-x)^{\frac{1}{2}+1}(-1)\right\} \\&=&-2x(a-x)^{\frac{1}{2}}+2\left\{\frac{-1}{\frac{3}{2}}(a-x)^{\frac{3}{2}}\right\} \\&=&-2x(a-x)^{\frac{1}{2}}+2\left\{\frac{-2}{3}(a-x)^{\frac{3}{2}}\right\} \\&=&-2x(a-x)^{\frac{1}{2}}-\frac{4}{3}(a-x)^{\frac{3}{2}} \\&=&-2x(a-x)^{\frac{1}{2}}-\frac{4}{3}(a-x)(a-x)^{\frac{1}{2}} \\&=&(a-x)^{\frac{1}{2}}\left\{-2x-\frac{4}{3}(a-x)\right\} \\&=&(a-x)^{\frac{1}{2}}\left(-2x-\frac{4}{3}a+\frac{4}{3}x\right) \\&=&(a-x)^{\frac{1}{2}}\left(-\frac{6}{3}x-\frac{4}{3}a+\frac{4}{3}x\right) \\&=&(a-x)^{\frac{1}{2}}\left(-\frac{2}{3}x-\frac{4}{3}a\right) \\&=&-\frac{2}{3}(a-x)^{\frac{1}{2}}\left(x+2a\right)+C\;\cdots\;C:積分定数 \\&=&-\frac{2}{3}\sqrt{a-x}\left(x+2a\right)+C \end{eqnarray} $$

定積分

$$ \begin{eqnarray} \int_{0}^{a} \frac{x}{\sqrt{a-x}}\mathrm{d}x &=&\left[-\frac{2}{3}\sqrt{a-x}\left(x+2a\right)\right]_{0}^{a} \\&=&(-\frac{2}{3}\sqrt{a-a}\left(a+2a\right))-(-\frac{2}{3}\sqrt{a-0}\left(0+2a\right)) \\&=&0-(-\frac{2}{3}\sqrt{a}\left(2a\right)) \\&=&0-(-\frac{4}{3}\sqrt{a}^3) \\&=&\frac{4}{3}a^{\frac{3}{2}} \end{eqnarray} $$