H2 Maths Notes (JC 1-2): 5.3) Integration Techniques

Before you revise\ Build a flowchart for integration: recognise the pattern (substitution, parts, partial fractions, trigonometric identities) before diving in. Always write differential substitutions explicitly.

Core Formulae

• Reverse of differentiation rules (power, exponential, trigonometric, logarithmic).
• Integration by substitution: choose \( u = g(x) \) to simplify integrand, rewrite \( dx = \frac{du}{g'(x)} \).
• Integration by parts: \( \int u \space dv = uv - \int v \space du \).
• Partial fractions: decompose rational functions into simpler fractions before integrating.

Substitution Examples

Example -- Trigonometric substitution

Evaluate \( \int \frac{2x}{x^2 + 1} \space dx \).

1. Let \( u = x^2 + 1 \) ⇒ \( du = 2x \space dx \).
2. Integral becomes \( \int \frac{1}{u} \space du = \ln\lvert u \rvert + C = \ln\lvert x^2 + 1 \rvert + C \).

Example -- Exponential substitution

Evaluate \( \int x e^{x^2} \space dx \).

1. Let \( u = x^2 \) ⇒ \( du = 2x \space dx \).
2. Integral \( = \tfrac{1}{2} \int e^{u} \space du = \tfrac{1}{2} e^{x^2} + C \).

Integration by Parts

• Choose \( u \) to simplify upon differentiation; set \( dv \) as remaining factor.
• For repeated parts, loop until integral becomes elementary.

Example -- \( x \cos x \)

Take \( u = x \space dv = \cos x \space dx \). \[ \int x \cos x \space dx = x \sin x - \int \sin x \space dx = x \sin x + \cos x + C. \]

Example -- Reduction formula

For \( I_n = \int x^n e^x \space dx \), set \( u = x^n \space dv = e^x \space dx \): \[ I_n = x^n e^x - n \int x^{n-1} e^x \space dx = x^n e^x - n I_{n-1}. \]

Partial Fractions

• Proper rational functions: express as sum of linear/quadratic factors.
• For repeated factors, include each power up to multiplicity.
• For irreducible quadratics, use ( \frac{Ax + B}{quadratic} ).

Example -- Partial fraction integral

Evaluate \( \int \frac{3x + 5}{x^2 - x - 2} \space dx \).

1. Factor denominator: \( (x - 2)(x + 1) \).
2. Decompose: \( \frac{3x + 5}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \).
3. Solve for \( A = 2 \space B = 1 \).
4. Integrate: \( \int \left( \frac{2}{x - 2} + \frac{1}{x + 1} \right) dx = 2 \ln\lvert x - 2 \rvert + \ln\lvert x + 1 \rvert + C \).

Trigonometric Identities

• Use \( \sin^2 x = \tfrac{1}{2}(1 - \cos 2x) \), \( \cos^2 x = \tfrac{1}{2}(1 + \cos 2x) \), and \( \sin x \cos x = \tfrac{1}{2} \sin 2x \).
• For integrals of \( \sec x \) or \( \tan x \), multiply numerator and denominator to match derivatives.

Example -- \( \int \sec^2 x \)

Recognise derivative of \( \tan x \: \) \( \int \sec^2 x \space dx = \tan x + C \).

Improper Integrals and Convergence

• Evaluate limits for infinite bounds or singularities: [ \int_1^\infty \frac{1}{x^p} \space dx \text{ converges iff } p > 1. ]
• For improper points inside interval, split integral and take limits.

Calculator Workflow

• Use GC to confirm antiderivatives by differentiating results.
• Numerical integration (∫f(x)dx) checks definite integrals; still show analytical steps.
• Store substitution results before back-substituting to reduce mistakes.

Exam Watch Points

• Explicitly state substitution and new limits for definite integrals.
• Include constant of integration for indefinite integrals.
• Handle absolute values in logarithms when integrating rational functions.
• Justify convergence when dealing with improper integrals.

Quick Revision Checklist

• [ ] Identify the right technique (substitution, parts, partial fractions) for each integrand.
• [ ] Carry out algebraic manipulations cleanly before integrating.
• [ ] Evaluate definite integrals with correct limits and substitution steps.
• [ ] Discuss convergence for improper integrals and document limit processes.

Next steps: Study Topic 5.4 for definite integrals and area/volume applications.