Integration Techniques — Syllabus Points

1. Reduction Formulae

• 1.1 Set up $I_n$ as a definite integral involving parameter $n$
• 1.2 Use integration by parts to derive a recurrence relation
• 1.3 State the recurrence in the form $I_n = f(n) \cdot I_{n-k} + g(n)$
• 1.4 Find explicit values for $I_0$ or $I_1$ as boundary conditions
• 1.5 Apply recurrence iteratively to evaluate $I_n$ for specific $n$

2. Integration by Parts

• 2.1 Apply $\int u\,dv = uv - \int v\,du$ with appropriate choice of $u$ and $dv$
• 2.2 Use LIATE rule (Log, Inverse trig, Algebraic, Trig, Exponential) to choose $u$
• 2.3 Apply repeated integration by parts when necessary

3. Integration by Substitution

• 3.1 Use trigonometric substitutions ($x = \sin\theta$, $x = \tan\theta$, etc.)
• 3.2 Use algebraic substitutions ($u = x^2$, $u = e^x$, etc.)
• 3.3 Change limits of definite integrals correctly

4. Integration of Rational Functions

• 4.1 Decompose proper fractions using partial fractions
• 4.2 Handle repeated linear factors: $\frac{A}{x-a}+\frac{B}{(x-a)^2}$
• 4.3 Handle quadratic factors: $\frac{Ax+B}{x^2+c}$
• 4.4 Integrate using standard forms: $\ln$, $\tan^{-1}$, etc.

5. Improper Integrals

• 5.1 Evaluate integrals with infinite limits
• 5.2 Evaluate integrals with integrand singularities