第4章 4.15

(1)

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

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

(Cは積分定数)

(2)

$\frac{x^3-x+4}{(x-1)(x-2)}$

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

$=\int x+3+\left(\frac{-4}{x-1}+\frac{10}{x-2}\right)dx = \frac{1}{2}x^2+3x+\log\frac{(x-2)^{10}}{(x-1)^4}+C$

(Cは積分定数)

(3)

$\frac{4x+19}{(x^2+4)(x+1)^2}$

$\int \frac{4x+19}{(x^2+4)(x+1)^2}dx = \int \left\{\frac{-2x-1}{(x^2+4)}+\frac{2x+5}{(x+1)^2}\right\}dx = \int \left\{\frac{-2x}{x^2+4}+\frac{-1}{x^2+4}+\frac{2x+2}{x^2+2x+1}+\frac{3}{(x+1)^2}\right\}dx$

$=-\frac{1}{2}\tan^{-1}\frac{x}{2}+\log\frac{(x+1)^2}{x^2+4}-\frac{3}{x+1}+C$

(Cは積分定数)

(4)

$\frac{x^2}{x^4-1}$

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

$=\int \frac{1}{4}\left(\frac{1}{x-1}+\frac{-1}{x+1}+\frac{2}{x^2+1}\right)dx = \frac{1}{2}\tan^{-1}x+\frac{1}{4}\log\left|\frac{x-1}{x+1}\right|+C$

(Cは積分定数)