**Candidates should be able to:**

- integrate hyperbolic functions and recognise integrals of functions of the form \(\frac{1}{\sqrt{a^2-x^2}}\), \(\frac{1}{\sqrt{x^2+a^2}}\) and \(\frac{1}{\sqrt{x^2-a^2}}\), and integrate associated functions using trigonometric or hyperbolic substitutions as appropriate (Including use of completing the square where necessary, e.g. to integrate \(\frac{1}{\sqrt{x^2+x}}\).)
- derive and use reduction formulae for the evaluation of definite integrals (e.g. \(\int_{0}^{\frac{1}{2}\pi}\sin^n x \ dx, \int_{0}^{1} e^{-x}(1 – x)^n dx\). In harder cases hints may be given, e.g. \(\int_{0}^{\frac{1}{4}\pi} \sec^n x \ dx\) by considering \(\frac{d}{dx} (\tan x \sec^n x)\).)
- understand how the area under a curve may be approximated by areas of rectangles, and use rectangles to estimate or set bounds for the area under a curve or to derive inequalities or limits concerning sums (Questions may involve either rectangles of unit width or rectangles whose width can tend to zero, e.g. \(1 + \ln n > \sum\limits_{r =1}^{n} \frac{1}{r} > \ln (n + 1) \ , \ \sum\limits_{r=1}^{n} \frac{1}{n} (1 + \frac{r}{n})^{-1} \approx \int_{0}^{1} (1 + x)^{-1} dx\).)
- use integration to find

–arc lengths for curves with equations in Cartesian coordinates, including the use of a parameter, or in polar coordinates

–surface areas of revolution about one of the axes for curves with equations in Cartesian coordinates, including the use of a parameter.

(Any questions involving integration may require techniques from Cambridge International A Level Mathematics (9709) applied to more difficult cases, e.g. integration by parts for \(\int e^x \sin x \ dx\), or use of the substitution \(t = \tan \frac{1}{2} x\). Surface areas of revolution for curves with equations in polar coordinates will not be required.)

**Integration – Further Pure Maths 2 Video (3 Parts)**

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