Evaluate the integral: I = integral dx/(x²+2x)

Evaluate the integral: $$ I = \int \frac{dx}{x^2+2x}. $$

Step 1 · Factor and set up partial fractions

$$ x^2+2x = x(x+2), \qquad \frac{1}{x(x+2)} = \frac Ax + \frac{B}{x+2}. $$

Multiplying by $x(x+2)$: $1=A(x+2)+Bx$. Setting $x=0$ gives $A=\tfrac12$; setting $x=-2$ gives $B=-\tfrac12$.

Step 2 · Integrate

$$ I = \int \left(\frac{1}{2x} - \frac{1}{2(x+2)}\right) dx = \frac12\ln|x| - \frac12\ln|x+2| + C. $$

Step 3 · Combine logarithms

$$ I = \frac12\ln\left|\frac{x}{x+2}\right| + C. $$

Summary Table

Additional Notes

• The domain excludes $x=0$ and $x=-2$, where the original integrand is undefined.
• This is the standard cover-up method for partial fractions with distinct linear factors: plugging in each root of the denominator isolates the corresponding coefficient directly.
• The result can be checked by differentiating $\tfrac12\ln|x|-\tfrac12\ln|x+2|$: this gives $\tfrac{1}{2x}-\tfrac{1}{2(x+2)}=\tfrac{(x+2)-x}{2x(x+2)}=\tfrac{1}{x(x+2)}$, matching the integrand.