第4章 4.6

(1)

$\dfrac{x-4}{(x-2)(x-3)}$

$\int \frac{x-4}{(x-2)(x-3)}dx = \int \frac{x-4}{x^2-5x+6}dx = \int\left\{\frac{1}{2}\cdot\frac{2x-5}{x^2-5x+6} + \frac{-\frac{3}{2}}{x^2-5x+6}\right\}dx$

$= \int\left\{\frac{1}{2}\cdot\frac{2x-5}{x^2-5x+6} + \frac{3}{2}\cdot\left(\frac{1}{x-2}-\frac{1}{x-3}\right)\right\}dx = \frac{1}{2}\log(x^2-5x+6) + \log\sqrt{\left(\frac{x-2}{x-3}\right)^3} + C$

$= \log\frac{(x-2)^2}{|x-3|} + C$

（C は積分定数）

(2)

$\dfrac{2x+7}{x^2+x-2}$

$\int \frac{2x+7}{x^2+x-2}dx = \int\left\{\frac{2x+1}{x^2+x-2}+\frac{6}{x^2+x-2}\right\}dx = \int\left\{\frac{2x+1}{x^2+x-2}+\frac{6}{(x+2)(x-1)}\right\}dx$

$= \int\left\{\frac{2x+1}{x^2+x-2}+2\cdot\left(\frac{1}{x-1}-\frac{1}{x+2}\right)\right\}dx = \log(x^2+x-2)+2\log\left|\frac{x-1}{x+2}\right|+C$

$= \log(x+2)(x-1)+\log\left(\frac{x-1}{x+2}\right)^2+C = \log\left|\frac{(x-1)^3}{x+2}\right|+C$

（C は積分定数）

(3)

$\dfrac{1}{x^3-x}$

$\int \frac{1}{x^3-x}dx = \int\frac{1}{x(x^2-1)}dx = \int\left(\frac{-1}{x}+\frac{\frac{1}{2}}{x-1}+\frac{\frac{1}{2}}{x+1}\right)dx = \frac{1}{2}\log\frac{|x^2-1|}{x^2}+C$

（C は積分定数）

(4)

$\dfrac{x^3+2x^2}{x^2-1}$

$\int \frac{x^3+2x^2}{x^2-1}dx = \int\left\{x+2+\frac{x+2}{x^2-1}\right\}dx = \int\left\{x+2+\frac{(x+1)+1}{(x-1)(x+1)}\right\}dx = \int\left\{x+2+\frac{1}{x-1}+\frac{1}{(x-1)(x+1)}\right\}$

$= \int\left\{x+2+\frac{1}{x-1}+\frac{1}{x^2-1}\right\}dx = \frac{1}{2}x^2+2x+\log(x-1)+\frac{1}{2}\log\left|\frac{(x-1)^3}{x+1}\right|+C$ （C は積分定数）

(5)

$\frac{2}{x(x^2+1)}$

$\int \frac{2}{x(x^2+1)}dx = \int 2\left\{\frac{1}{x}-\frac{x}{x^2+1}\right\}dx = 2\log x - \log(x^2+1)+C = \log\frac{x^2}{x^2+1}+C$ （C は積分定数）

(6)

$\frac{x-2}{x^3+4x}$

$\int \frac{x-2}{x^3+4x}dx = \int \frac{x-2}{x(x^2+4)}dx = \int \frac{1}{2}\left\{\frac{-1}{x}+\frac{x+2}{x^2+4}\right\}dx = \int \frac{1}{2}\left\{-\frac{1}{x}+\frac{2x}{x^2+4}\cdot\frac{1}{2}+\frac{2}{x^2+4}\right\}dx$

$= \frac{1}{2}\left\{-\log x+\frac{1}{2}\log(x^2+4)+\tan^{-1}\frac{x}{2}\right\}+C = \frac{1}{4}\log\frac{x^2+4}{x^2}+\frac{1}{2}\tan^{-1}\frac{x}{2}+C$ （C は積分定数）