Integration by parts: formulas, proof, and four useful patterns
Derive integration by parts from the product rule and practice logarithmic, polynomial-trigonometric, polynomial-exponential, and cyclic integrals.
Logarithmic differentiation: variable powers, products, and worked problems
Differentiate f(x) raised to g(x), handle absolute values and domain restrictions, and solve minimum, exponential-bound, and root-count problems.
How extra calculus and complex numbers can help humanities mathematics
Limits as checks, derivatives for optimization, and complex multiplication for rotations: four benefits and worked examples beyond a basic mathematics course.
Circle inversion in the complex plane: properties and worked problems
Understand w = 1/conjugate(z), prove how circles and lines transform, and solve three locus problems with complete domain checks.
Three useful limit formulas, their meaning, and how to derive related limits
Connect sine and exponential limits with local graph behavior, derive logarithmic and cosine limits, and keep the logical starting point of a proof clear.
Areas enclosed by parametric curves: cardioids, astroids, and a check
Set up signed integrals with the correct branches, calculate two complete examples, and verify them using the parametric area formula.
The limits of sin x/x at zero and infinity, with a noncircular proof
Use the squeeze theorem for both limits, distinguish radians from degrees, and justify the sector-area argument without assuming trigonometric derivatives.