Calculus
Extension 2 integration techniques: method selection, integration by parts, reduction formulae, partial fractions, t-substitution (Weierstrass), trigonometric identities, completing the square, and partial fractions with irreducible quadratic factors.
Active lessons
9
Status
Active
Lesson flow
Learn -> Guided Practice -> Independent Practice -> Mastery Quiz
How to use this unit
- Start from lesson 1 and move through the pathway in order.
- Use guided practice before attempting independent practice.
- Mastery quizzes check whether the skill is ready for review or extension.
Unit pathway
Lesson pathway
9 of 9 active
Lesson 1
Advanced Integration Method Selection
Identify which technique to use before computing: substitution, integration by parts, partial fractions, standard form, or trigonometric identities.
Lesson 2
Integration by Parts Extension
Apply integration by parts to repeated products, logarithmic integrands, and definite integrals with exact answers.
Lesson 3
Reduction Formulae Introduction
Use supplied reduction formulae to evaluate families of integrals recursively from base cases.
Lesson 4
Partial Fractions Integration
Decompose proper rational functions into partial fractions over distinct or repeated linear factors, then integrate each term to produce logarithmic or power expressions.
Lesson 5
t-Substitution (Weierstrass)
Apply the Weierstrass substitution t = tan(x/2) to convert integrands involving sin x and cos x into rational functions of t, then integrate using standard techniques.
Lesson 6
Integration Using Trigonometric Identities
Use half-angle identities sin²x = (1−cos2x)/2 and cos²x = (1+cos2x)/2 to reduce powers of sine and cosine before integrating, and handle mixed products using product-to-sum identities.
Lesson 7
Integration by Completing the Square
Complete the square on a quadratic denominator and use the standard arctan form ∫dx/(x²+a²) = (1/a)arctan(x/a)+C to evaluate integrals of rational functions.
Lesson 8
Volumes of Revolution
Use the disk and washer methods to find volumes formed by rotating regions about the x-axis and y-axis: V = π∫[f(x)]² dx and V = π∫([f(x)]²−[g(x)]²) dx.
Lesson 9
Partial Fractions with Quadratic Factors
Decompose rational functions containing irreducible quadratic factors into partial fractions, and use polynomial long division when the degree of the numerator is not less than the degree of the denominator.
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