Examples and practice
Techniques of integration
Back to lessonPractice section 1 · Questions 1–6
Reverse chain rules, powers, logarithms and exponentials
Practice section 2 · Questions 7–12
Exponential, inverse-trigonometric and logarithmic standard forms
Practice section 3 · Questions 13–18
Partial fractions, denominator derivatives and trigonometric composites
Practice section 4 · Questions 19–24
Trigonometric identities and substitution
Practice section 5 · Questions 25–30
Integration by parts, given derivatives and direct integrals
Practice section 6 · Questions 31–36
Mixed methods: standard forms, partial fractions, trigonometry, substitution and parts
Practice section 7 · Questions 37–42
Combining given derivatives, trigonometric identities and parts
Practice section 8 · Questions 43–48
Substitution, trigonometric identities and derivative-assisted parts
- Expand after the specified linear substitution
- Use a shown antiderivative inside integration by parts
- Transform a square-root integral and reverse an exponential chain
- Apply double-angle and product-to-sum identities before integrating
- Complete a square after substitution and use a product derivative
- Use the prescribed cosine substitution in a difference of squares
Practice section 9 · Questions 49–54
Choosing between standard forms, partial fractions and repeated parts
- Reverse an arctangent chain and use a principal sine substitution
- Choose among inner derivatives, completed squares and parts
- Split a numerator into a denominator derivative and a remainder
- Split the numerator and complete the square under a radical
- Find partial fractions and integrate each term
- Solve a returning integration-by-parts equation in two ways
Practice section 10 · Question 55
Compare integration by parts with an arctangent substitution