Question

Integration by Parts Fractions

1.) ∫dy/(y^2+2y)

2.) ∫(x^3+x+2)/(x^2-1)dx

Answer 1

1)

$$\int\dfrac{dy}{y^2+2y}$$

$$\dfrac{1}{y^2+2y}=\dfrac{A}{y}+\dfrac{B}{y+2}=\dfrac{A(y+2)+By}{y(y+2)}$$

$y=0: 2A=1=>A=\dfrac{1}{2}$

$y=-2: -2B=1=>B=-\dfrac{1}{2}$

$$\int\dfrac{dy}{y^2+2y}=\dfrac{1}{2}\int\dfrac{dy}{y}-\dfrac{1}{2}\int\dfrac{dy}{y+2}$$

$$=\dfrac{1}{2}\ln|y|-\dfrac{1}{2}\ln|y-2|+C$$

2)

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

$$=\int xdx+\int\dfrac{2(x+1)}{(x+1)(x-1)}dx$$

$$=\dfrac{x^2}{2}+2\int\dfrac{dx}{x-1}=\dfrac{x^2}{2}+2\ln|x-1|+C$$