multivariable_08_Techniques_of_Integration_2up

multivariable_08_Techniques_of_Integration_2up - 8...

Info iconThis preview shows pages 1–3. 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: 8 Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. For example, faced with integraldisplay x 10 dx we realize immediately that the derivative of x 11 will supply an x 10 : ( x 11 ) = 11 x 10 . We dont want the 11, but constants are easy to alter, because differentiation ignores them in certain circumstances, so d dx 1 11 x 11 = 1 11 11 x 10 = x 10 . From our knowledge of derivatives, we can immediately write down a number of an- tiderivatives. Here is a list of those most often used: integraldisplay x n dx = x n +1 n + 1 + C, if n negationslash = 1 integraldisplay x 1 dx = ln | x | + C integraldisplay e x dx = e x + C integraldisplay sin xdx = cos x + C 149 150 Chapter 8 Techniques of Integration integraldisplay cos xdx = sin x + C integraldisplay sec 2 xdx = tan x + C integraldisplay sec x tan xdx = sec x + C integraldisplay 1 1 + x 2 dx = arctan x + C integraldisplay 1 1 x 2 dx = arcsin x + C 8b i i Needless to say, most problems we encounter will not be so simple. Heres a slightly more complicated example: find integraldisplay 2 x cos( x 2 ) dx. This is not a simple derivative, but a little thought reveals that it must have come from an application of the chain rule. Multiplied on the outside is 2 x , which is the derivative of the inside function x 2 . Checking: d dx sin( x 2 ) = cos( x 2 ) d dx x 2 = 2 x cos( x 2 ) , so integraldisplay 2 x cos( x 2 ) dx = sin( x 2 ) + C. Even when the chain rule has produced a certain derivative, it is not always easy to see. Consider this problem: integraldisplay x 3 radicalbig 1 x 2 dx. There are two factors in this expression, x 3 and radicalbig 1 x 2 , but it is not apparent that the chain rule is involved. Some clever rearrangement reveals that it is: integraldisplay x 3 radicalbig 1 x 2 dx = integraldisplay ( 2 x ) parenleftbigg 1 2 parenrightbigg (1 (1 x 2 )) radicalbig 1 x 2 dx. This looks messy, but we do now have something that looks like the result of the chain rule: the function 1 x 2 has been substituted into (1 / 2)(1 x ) x , and the derivative 8.1 Substitution 151 of 1 x 2 , 2 x , multiplied on the outside. If we can find a function F ( x ) whose derivative is (1 / 2)(1 x ) x well be done, since then d dx F (1 x 2 ) = 2 xF (1 x 2 ) = ( 2 x ) parenleftbigg 1 2 parenrightbigg (1 (1 x 2 )) radicalbig 1 x 2 = x 3 radicalbig 1 x 2 But this isnt hard: integraldisplay 1 2 (1 x ) xdx = integraldisplay 1 2 ( x 1 / 2 x 3 / 2 ) dx (8 . 1) = 1 2 parenleftbigg 2 3 x 3 / 2 2 5 x 5 / 2 parenrightbigg + C = parenleftbigg 1 5 x 1 3 parenrightbigg...
View Full Document

Page1 / 10

multivariable_08_Techniques_of_Integration_2up - 8...

This preview shows document pages 1 - 3. Sign up to view the full document.

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