Splitting the Numerator
Splitting the Numerator
\[\mathrm{f}(x)=A\mathrm{g}'(x)+B\]
\[\int\frac{\mathrm{f}(x)}{\mathrm{g}(x)}\,\mathrm{d}x=A\int\frac{\mathrm{g}'(x)}{\mathrm{g}(x)}\,\mathrm{d}x+B\int\frac{1}{\mathrm{g}(x)}\,\mathrm{d}x\]
\[\int\frac{\mathrm{f}(x)}{\sqrt{\mathrm{g}(x)}}\,\mathrm{d}x=A\int\frac{\mathrm{g}'(x)}{\sqrt{\mathrm{g}(x)}}\,\mathrm{d}x+B\int\frac{1}{\sqrt{\mathrm{g}(x)}}\,\mathrm{d}x\]
| Integrand | Recognisable term | Remaining standard term |
|---|---|---|
| \(\dfrac{\mathrm{f}(x)}{\mathrm{g}(x)}\) | \(A\dfrac{\mathrm{g}'(x)}{\mathrm{g}(x)}\) | \(\dfrac{B}{\mathrm{g}(x)}\) |
| \(\dfrac{\mathrm{f}(x)}{\sqrt{\mathrm{g}(x)}}\) | \(A\dfrac{\mathrm{g}'(x)}{\sqrt{\mathrm{g}(x)}}\) | \(\dfrac{B}{\sqrt{\mathrm{g}(x)}}\) |
Worked pattern
\(8x+3=(8x+2)+1\)
\(\int\frac{8x+3}{4x^2+2x+1}\,\mathrm dx=\int\frac{8x+2}{4x^2+2x+1}\,\mathrm dx+\int\frac1{4x^2+2x+1}\,\mathrm dx\)
\(u=4x^2+2x+1\Rightarrow\mathrm du=(8x+2)\,\mathrm dx\)
\(4x^2+2x+1=4\left[\left(x+\frac14\right)^2+\left(\frac{\sqrt3}4\right)^2\right]\)
\(=\ln|4x^2+2x+1|+\frac1{\sqrt3}\tan^{-1}\left(\frac{4x+1}{\sqrt3}\right)+C\)
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