FP3 · Chapter 4
Integration
Every integral in this chapter is really the same three moves in disguise: spot a pattern you already know, force the integrand into that shape, then execute — whether that's a clean substitution, a "reverse chain rule" guess, or building a reduction formula for when the power just won't quit.
📎 Builds on: Chapter 1 (Hyperbolic Functions) — you'll lean heavily on sinh, cosh, their derivatives, and the log forms of arsinh/arcosh/artanh. Chapter 3's derivatives of inverse functions also feed straight into the standard substitutions here.
Summary
- Recognise a standard function — some integrals are just derivatives you already know, run backwards.
- Using formulae to change the integrand — trig/hyperbolic identities (like double angle formulae) turn an unintegrable expression into one you can do term by term.
- Reverse chain rule — spot the "function of a function times its derivative" pattern, guess the antiderivative shape, differentiate to fix the constant.
- Standard substitutions — four go-to substitutions (trig or hyperbolic) for integrals containing a² ± b²x², with or without a square root.
- Integrating inverse functions and ln x — integration by parts with dv/dx = 1, a trick that works because you can differentiate the inverse function even though you can't easily integrate it.
- Reduction formulae — for integrals with a power n, express Iₙ in terms of Iₙ₋₁ or Iₙ₋₂ instead of integrating from scratch every time.
- Arc length — using Pythagoras on an infinitesimal triangle to find the length of a curve, in Cartesian or parametric form.
- Area of a surface of revolution — spinning a curve around the x-axis and adding up the surface area of infinitely many thin cylindrical bands.
Standard Techniques
Before reaching for anything clever, always ask: "Have I seen this exact shape before, as a derivative?" This section builds up your toolkit in the order you should actually try it in an exam, from laziest to most involved.
1. Recognise a standard function
This is the "if it's staring you in the face, just write the answer down" category. If you can look at the integrand and immediately think "that's the derivative of something," you're done — no working required beyond stating the antiderivative and checking it by differentiating.
Recognise & reverse
∫ sec x tan x dx = sec x + c ∫ sech x tanh x dx = −sech x + c
These work because d(sec x)/dx = sec x tan x and d(sech x)/dx = −sech x tanh x — you memorised these derivatives in Chapter 3, and integration is just running the film backwards.
Worked Example
Find ∫ cosech x coth x dx.
Recall d(cosech x)/dx = −cosech x coth x.
So ∫ cosech x coth x dx = −cosech x + c
✏️ Practice
Find ∫ sech²x dx.
2. Using formulae to change the integrand
Sometimes there's no standard derivative that matches — yet. But an identity (Pythagorean, double angle, Osborne's rule) can rewrite the integrand into a form that does match. This is the "disguise removal" technique.
🧠 Why this works
You're not changing the value of the integral — you're rewriting the same function using an identity so that what you're integrating suddenly looks like something in your standard list.
Worked Example
tan x − x + c
Worked Example (hyperbolic version)
½(½ sinh 2x − x) + c
💡 Osborne's rule reminder
product of two sines
✏️ Practice