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