Choosing an Integration Technique

Power rule and standard integrals: SL 5.5 and SL 5.10.

\(\int\dfrac{\mathrm{f}(x)}{\mathrm{g}(x)}\,\mathrm{d}x\quad\text{or}\quad\int\dfrac{\mathrm{f}(x)}{\sqrt{\mathrm{g}(x)}}\,\mathrm{d}x\)

1. Do long division if the degree of \(\mathrm{f}(x)\) is at least the degree of \(\mathrm{g}(x)\).
2. If \(\mathrm{f}(x)\) is constant and \(\mathrm{g}(x)\) is quadratic, complete the square. Use the integrals of
   \(\dfrac{1}{a^2+x^2},\qquad\dfrac{1}{\sqrt{a^2-x^2}}\) in AA Formula Booklet AHL 5.15.
3. If \(\mathrm{g}(x)\) factorises into at most two distinct linear factors, express \(\dfrac{\mathrm{f}(x)}{\mathrm{g}(x)}\) in partial fractions and integrate.
4. Lastly, consider “Split Up Method”. When applicable, let \(\mathrm{f}(x)=A[\mathrm{g}'(x)]+B\), solve for \(A\) and \(B\), then integrate:
   \(\int\dfrac{\mathrm{f}(x)}{\mathrm{g}(x)}\,\mathrm{d}x=\int\Biggl(\underbrace{\dfrac{A\mathrm{g}'(x)}{\mathrm{g}(x)}}_{\substack{\text{Cancelling}\\\text{Method}}}+\underbrace{\dfrac{B}{\mathrm{g}(x)}}_{\substack{\text{Standard}\\\text{form}}}\Biggr)\mathrm{d}x\)
   or
   \(\int\dfrac{\mathrm{f}(x)}{\sqrt{\mathrm{g}(x)}}\,\mathrm{d}x=\int\Biggl(\underbrace{\dfrac{A\mathrm{g}'(x)}{\sqrt{\mathrm{g}(x)}}}_{\substack{\text{Cancelling}\\\text{Method}}}+\underbrace{\dfrac{B}{\sqrt{\mathrm{g}(x)}}}_{\substack{\text{Standard}\\\text{form}}}\Biggr)\mathrm{d}x\)

\(\int u\dfrac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x=uv-\int v\dfrac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x\) [AA Formula Booklet AHL 5.16]

• Choose \(u\) to simplify on differentiation; integrate \(\mathrm{d}v\). Repeat if needed.