第8章 8.5

(1)

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

(2)

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

(3)

$\int \frac{t^3}{t^2-3t+2}dt = \int \left(t+3+\frac{7t-6}{t^2-3t+2}\right)dt = \int \left(t+3+\frac{7t-6}{(t-1)(t-2)}\right)dt = \int \left(t+3+\frac{-1}{t-1}+\frac{8}{t-2}\right)dx$

$= \frac{1}{2}t^2+3t-\log(t-1)+8\log(t-2)+C = \frac{1}{2}t^2+3t+\log\frac{(t-2)^8}{|t-1|}+C$

(4)

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

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