Splitting the Numerator

For a linear numerator over a quadratic \(\mathrm{g}(x)\), write the numerator as \(A\mathrm{g}\prime(x)+B\). Match coefficients to find \(A\) and \(B\).

\(\int\dfrac{A\mathrm{g}\prime(x)+B}{\mathrm{g}(x)}\,\mathrm{d}x=A\ln|\mathrm{g}(x)|+B\int\dfrac{1}{\mathrm{g}(x)}\,\mathrm{d}x\)

\(\int\dfrac{A\mathrm{g}\prime(x)+B}{\sqrt{\mathrm{g}(x)}}\,\mathrm{d}x=2A\sqrt{\mathrm{g}(x)}+B\int\dfrac{1}{\sqrt{\mathrm{g}(x)}}\,\mathrm{d}x\)

Complete the square or factorise to handle the remaining integral. Check the real domain before using a square root or logarithm.

\(\begin{aligned}8x+3&=(8x+2)+1\\4x^2+2x+1&=\dfrac{(4x+1)^2+3}{4}\end{aligned}\)

\(\int\dfrac{8x+3}{4x^2+2x+1}\,\mathrm{d}x=\ln(4x^2+2x+1)+\dfrac{1}{\sqrt3}\arctan\left(\dfrac{4x+1}{\sqrt3}\right)+C\)