Question

integrate ((x+1)/(x^2+2x))dx

Answer 1

Partial fractions:
\[\frac{x+1}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2} \implies Ax+2A+Bx=x+1 \implies A=\frac{1}{2}, B=\frac{1}{2}\]
\[\int\limits \frac{x+1}{x^2+2x}dx=\frac{1}{2} \int\limits \frac{dx}{x}+\frac{1}{2} \int\limits \frac{dx}{x+2}=\frac{1}{2}\ln|x|+\frac{1}{2}\ln|x+2|+C\]
\[=\ln|\sqrt{x}*\sqrt{x+2}|+C=\ln|\sqrt{x^2+2x}|+C\]