# Integration – Further Pure Maths 2

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)