Partial fractions

Problem

Compute $\displaystyle\int_3^5\frac{2x+3}{5(x-1)(x+2)}\,dx$ with partial fractions.

Solution

The denominator is already factorised. We decompose into $\dfrac{C_1}{x-1}+\dfrac{C_2}{x+2}$; with the “cover-up” trick: $C_1 = \frac{2\cdot 1+3}{5(1+2)} = \frac{1}{3}, \qquad C_2 = \frac{2(-2)+3}{5(-2-1)} = \frac{1}{15}.$ Hence $\int_3^5\frac{2x+3}{5(x-1)(x+2)}\,dx = \left[\frac{1}{3}\ln|x-1|+\frac{1}{15}\ln|x+2|\right]_3^5.$ Evaluating: $\left(\tfrac13\ln 4+\tfrac1{15}\ln 7\right)-\left(\tfrac13\ln 2+\tfrac1{15}\ln 5\right) = \boxed{\tfrac13\ln 2 + \tfrac1{15}\ln\tfrac{7}{5}}\approx 0{,}254.$

Links

Topics: Integral
Concepts: Partial fractions · Definite integral
Methods: Integration by partial fractions
Skills: Integrate · Factorise
Exercise types: Integral computation