部分分数展開

1. $$I=\int \frac{x^2+x+1}{x+2}dx =\int\left\{x-1+\frac{3}{x+2}\right\}dx$$
2. $$I=\int \frac{1}{(x+3)(x+4)}dx =\int\left\{\frac{A}{x+3}+\frac{B}{x+4}\right\}dx$$
   $$\begin{eqnarray*} A&=& \left.\frac{1}{x+4}\right\vert _{x=-3}=1\\ B&=& \left.\frac{1}{x+3}\right\vert _{x=-4}=-1 \end{eqnarray*}$$
   
   

3. $$I=\int \frac{x-7}{x^2-2x-3}dx =\int\left\{\frac{A}{x+1}+\frac{B}{x-3}\right\}dx$$
   $$\begin{eqnarray*} A&=& \left.\frac{x-7}{x-3}\right\vert _{x=-1}\\ B&=& \left.\frac{x-7}{x+1}\right\vert _{x=3} \end{eqnarray*}$$
   
   

4. $$I=\int \frac{2x+3}{x(x+2)(x+3)}dx$$
   
   $$\frac{2x+3}{x(x+2)(x+3)} =\frac{a}{x}+\frac{b}{x+2}+\frac{c}{x+3}$$
   
   
   $$\begin{eqnarray*} a&=& \left.\frac{2x+3}{(x+2)(x+3)}\right\vert _{x=0}\\ b&=&... ...t _{x=-2}\\ c&=& \left.\frac{2x+3}{x(x+2)}\right\vert _{x=-3} \end{eqnarray*}$$