Integration - CALCULUS 133 TECHNIQUES OF INTEGRATION The purpose of these techniques is the following you are given the problem of finding an

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CALCULUS 133: TECHNIQUES OF INTEGRATION The purpose of these techniques is the following: you are given the problem of finding an antiderivative of a complicated function, and these techniques allow you to reduce it to finding an antiderivative of a simpler function. Eventually you reduce the problem to finding an antiderivative of a “standard function”, for instance sin or cos or e x . A list of standard functions, for which we know the antiderivatives is given on page 383. I expect you to know 1,2,3,5,6,11, 13 and 14. 1. Integration by substitution This technique involves introducing a new variable u for x , which will hopefully simplify the integral. We let u = f ( x ) be some function of x and then du = f ( x ) dx . Then substitute in u and du in your integral. You have to be clever to make sure all the x ’s cancel. Be careful when you integrate definite integrals to change the bounds. Example To find R e x 1+ e x dx , let u = 1 + e x so that du = e x dx . Then Z e x 1 + e x dx = Z 1 u du = ln | u | + C = ln(1 + e x ) + C. Example To find R π/ 2 cos x 1+sin 2 x dx , let u = sin x so that du = cos x dx . Then when x = 0, u = 0 and when x = π/ 2, u = 1. Therefore, Z π/ 2 cos x 1 + sin 2 x dx = Z 1 1 1 + u 2 du = tan- 1 ( u ) 1 = tan- 1 (1)- tan- 1 (0) = √ 2 2- 0 = √ 2 2 . Example To find R tan x dx = R sin x cos x dx , let u = cos x , then du =- sin x dx . Therefore, Z sin x cos x dx =- Z 1 u du =- ln | u | + C =- ln | cos x | + C. 2. Trigonometric Integrals To compute R sin n x dx , R cos n x dx or R sin m x cos n x dx , where m and n are positive integers, the following ideas are handy. If sinpositive integers, the following ideas are handy....
View Full Document

This note was uploaded on 07/14/2011 for the course MAC 3473 taught by Professor Block during the Spring '08 term at University of Florida.

Page1 / 5

Integration - CALCULUS 133 TECHNIQUES OF INTEGRATION The purpose of these techniques is the following you are given the problem of finding an

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online