Advanced Integration
Choose an integration strategy: substitution, trigonometric substitution, \(t=\tan x\), powers of sine/cosine, and integration by parts.
1. Choosing a method
Advanced integration is less about memorising many rules and more about recognising structure. Before integrating, ask:
- Is there a composite function and its derivative? Try substitution.
- Is there a square root such as \(\sqrt{a^2-x^2}\), \(\sqrt{a^2+x^2}\), or \(\sqrt{x^2-a^2}\)? Consider trigonometric substitution.
- Does the integral contain rational expressions in \(\sin x\) and \(\cos x\)? Consider \(t=\tan\frac{x}{2}\) or another tangent substitution, depending on the syllabus/teacher convention.
- Is it a product such as polynomial times exponential/trig/log? Consider integration by parts.
2. Substitution with a radical
Evaluate \(\int x\sqrt{x^2+1}\,dx\).
Let \(u=x^2+1\). Then \(du=2x\,dx\), so \(x\,dx=\frac12du\).
\[\int x\sqrt{x^2+1}\,dx=\frac12\int u^{1/2}\,du=\frac13u^{3/2}+C=\frac13(x^2+1)^{3/2}+C\]
3. Integration by parts reminder
\[\int u\,dv=uv-\int v\,du\]
Choose \(u\) to become simpler when differentiated. Common choices for \(u\) include logarithmic functions, inverse trig functions, and polynomials.