Mathematics · University
Integral methods and improper limits · revision notes
Substitution reverses a chain rule; integration by parts reverses a product rule.
Key ideas and conditions
Choose a method
Substitution reverses a chain rule; integration by parts reverses a product rule.
Endpoint limits
An improper integral is a limit of proper integrals. Check every singular endpoint separately.
Convergence test
For x ≥ 1, the integral of x^(−p) converges exactly when p > 1.
Common trap
A finite antiderivative expression does not establish convergence until the boundary limits are evaluated.
Explain without looking
- Show that the integral of exp(−ax) from 0 to infinity equals 1/a when a>0.
- In substitution u=x² for a definite integral, what must be changed besides the differential?
Further reading and review references
These are reference links, not copied exercises, institutional endorsement or proof of independent review.
AI-assisted preview; independent educational review required. Level labels describe prerequisites, not accreditation or official curriculum alignment.