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LearnEdexcelYFM012018 specificationWFM03 · Further Pure Mathematics 3Integration

WFM03 · Module

Integration

2 subtopics≈ 95 min in total

Start the module: Inverse trig and hyperbolic functions in calculus Open graphing calculator

Inverse trig and hyperbolic functions in calculus
Differentiating $y=\arcsin x$ implicitly gives $\frac1{\sqrt{1-x^2}}$ ; similar results hold for $\arctan$ , $\operatorname{arsinh}$ , $\operatorname{arcosh}$ . Read backwards, these give the integrals of $\frac1{\sqrt{a^2-x^2}}$ , $\frac1{a^2+x^2}$ and $\frac1{\sqrt{x^2\pm a^2}}$ , which algebra alone can't do.
≈ 45 min
Reduction formulae, arc length and surface area
A reduction formula links $I_n=\int f_n$ to $I_{n-1}$ or $I_{n-2}$ (usually by parts), so you climb from an easy $I_0$ to any $I_n$ . The length of a curve is $\int\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$ and the surface it sweeps out is $2\pi\int y\,ds$ .
≈ 50 min