Evaluate the integral: I = integral dx/(x²+5x)

Evaluate the integral: $$ I = \int \frac{dx}{x^2+5x}. $$

Step 1 · Factor and set up partial fractions

$$ x^2+5x = x(x+5), \qquad \frac{1}{x(x+5)} = \frac Ax + \frac{B}{x+5}. $$

Multiplying through: $1=A(x+5)+Bx$. Setting $x=0$: $A=\tfrac15$. Setting $x=-5$: $B=-\tfrac15$.

Step 2 · Integrate

$$ I = \frac15\int \frac{dx}{x} - \frac15\int \frac{dx}{x+5} = \frac15\ln|x| - \frac15\ln|x+5| + C. $$

Step 3 · Combine logarithms

$$ I = \frac15\ln\left|\frac{x}{x+5}\right| + C. $$

Summary Table

Additional Notes

• The domain excludes $x=0$ and $x=-5$, the roots of the denominator.
• This follows the identical pattern to Problems 187 and 188 elsewhere in this collection, with the second root shifted to $-5$; in general, $\int dx/[x(x+k)]=\tfrac1k\ln|x/(x+k)|+C$.
• The result can be checked by differentiating $\tfrac15\ln|x|-\tfrac15\ln|x+5|$: this gives $\tfrac{1}{5x}-\tfrac{1}{5(x+5)}=\tfrac{(x+5)-x}{5x(x+5)}=\tfrac{1}{x(x+5)}$, matching the integrand.