This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Lecture 3:: Trigonometric Substitutions Revisited 0.1 Reduction Formulae: One idea which is useful is that of a reduction formula. The basic idea is to reduce an integral to a similar integral of lower order. For example: to compute R sin n ( x ) we might do the following Z sin n ( x ) dx =- cos( x ) sin n- 1 ( x ) + ( n- 1) Z cos 2 ( x ) sin n- 1 ( x ) dx n Z sin n ( x ) dx =- cos( x ) sin n- 1 ( x ) + ( n- 1) Z sin n- 1 ( x ) dx Z sin n ( x ) dx =- 1 n cos( x ) sin n- 1 ( x ) + n- 1 n Z sin n- 2 ( x ) dx Thus we can relate R sin 2 ( x ) dx to R sin ( x ) dx = R dx , R sin 4 ( x ) dx to R sin 2 ( x ) dx. Similarly for odd powers we find that we can relate R sin 3 ( x ) dx to R sin( x ) dx , R sin 5 ( x ) dx to R sin 3 ( x ) Self Study Problems: Compute R sin 6 ( x ) dx using the above reduction formula. Derive an analogous reduction formula for R cos n ( x ) dx . 0.2 General Trig Substitutions: To start with I will calculate a couple of integrals well use a bunch of times in...
View Full Document