有理関数の積分法(integration of rational functions)

確認問題

1.: 次の関数を部分分数分解しよう．
(a) $\displaystyle{f(x) = \frac{7}{(x-2)(x+5)} }$ (b) $\displaystyle{f(x) = \frac{x^2 + 1}{x(x^2 - 1)}}$ (c) $\displaystyle{f(x) = \frac{x^2 + 3}{x^2 - 3x + 2}}$ (d) $\displaystyle{f(x) = \frac{x^2}{(x - 1)^2(x + 1)}}$ (e) $\displaystyle{f(x) = \frac{x^{5}}{(x-2)^2}}$ (f) $\displaystyle{f(x) = \frac{x+1}{x(x^{2} + 1)}}$

演習問題

1.

次の積分を求めよう．

(a) $\displaystyle{\int{\frac{7}{(x-2)(x+5)}} dx}$ (b) $\displaystyle{\int{\frac{x^2 + 1}{x(x^2 - 1)}} dx}$ (c) $\displaystyle{\int{\frac{x^2 + 3}{x^2 - 3x + 2}} dx}$

(d) $\displaystyle{\int{\frac{x^2}{(x - 1)^2(x + 1)}} dx}$ (e) $\displaystyle{\int{\frac{dx}{(x^2 + 16)^2}}}$ (f) $\displaystyle{\int{\frac{x^{5}}{(x-2)^2}} dx}$

(g) $\displaystyle{\int{\frac{x^5}{x^9 - 1}} dx (x^3 = t)}$ (h) $\displaystyle{\int{\frac{dx}{x(x^4 + 1)}} (x^4 = t)}$