partielle-integration-brüche-hü

6.54) a) nur detailierter lösungsweg, wegen edi in der stunde $\frac{5x-1}{x^2-1}$ $x^2-1=0\to x_1=-1;x_2=1$ x^2-1=(x+1)*(x-1)$$$$\frac{A}{x-1}+\frac{B}{x-1}=\frac{5x-1}{x^2-1}→5x-1=A(x-1)+B(x+1) $|:5\ast (-1)-1=A((-1)-1)+B((-1)+1)\to 6=-2A$ $||:5\ast 1=1=A(1-1)+B(1+1)4=2B$ A=3;B=2 $\to \frac{5x-1}{x^2-1}=\frac{3}{x+1}+\frac{2}{x-1}$ c) $\frac{3x^2+6x+2}{x^3+3x^2+2x}=\frac{3x^2+6x+2}{x(x^2+3x+2)}$ $\frac{A}{x}=\frac{B}{x+1}=\frac{C}{x+2}$ $\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2}$ 6.55) a) b) 6.56) a) $\frac{x+12}{x^2-x-6}=\frac{3}{x-3}=\frac{2}{x+2}$ $\int \frac{x+12}{x^2-x-6}dx=\ln |x-3|-2\ln |x+2|+C$ b) $\frac{7x-2}{2x^2+x-1}=\frac{1}{2x-1}+\frac{3}{x+1}$ $\int \frac{7x-2}{2x^2+x-1}dx=\frac{1}{2}\ln |2x-1|+3\ln |x+1|+c$ c) $\frac{3x^2-2x+1}{x^3-x}=-\frac{1}{x}+\frac{4x-2}{x^2-1}$ $\int \frac{3x^2-2x+1}{x^3-x}dx=-\ln |x|+2\ln |x^2-1|+C$ 6.57)