105-1-AntiderivativesSubstitution

105-1-AntiderivativesSubstitution - 1 Antiderivatives...

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: 1 Antiderivatives & Substitution 1.1 Antiderivatives Reminder about Derivatives First Ill remind you of a few derivatives that well be using a lot: ( x n ) = nx n- 1 , (sin x ) = cos x, (cos x ) =- sin x, (tan x ) = sec 2 x, (sec x ) = sec x tan x, (sin- 1 x ) = 1 1- x 2 , (tan- 1 x ) = 1 x 2 + 1 , ( e x ) = e x , (ln | x | ) = 1 x About (ln | x | ) = 1 x The last one deserves some extra attention, since this will show up again and again in integration. You may only remember the derivative (ln x ) = 1 x ; this is of course correct, but is only true for x in the domain of ln x , so for x > 0. For x 0, ln x doesnt exist, and so a derivative wouldnt make sense. We can (and should) extend this differentiation formula to negative x by taking ln | x | instead. Think of the graphs: ln x only has a graph to the right of the y-axis, while ln | x | also has the mirror image of the same graph to the left of the y-axis. We do have to actually check that the derivative is still 1 /x for x < 0, which we do as follows: x < | x | =- x (ln | x | ) = (ln(- x )) = 1- x (- 1) = 1 x . Antidifferentiation Antidifferentiation (or taking the derivative) is doing differentiation in reverse: given a function f ( x ), find a function F ( x ) such that F ( x ) = f ( x ). This F ( x ) is then called an antiderivative of f ( x ). Here are a few examples of important antiderivatives that you can obtain by reversing the differentiation formulas above: f ( x ) = x n- F ( x ) = 1 n + 1 x n +1 ( for n 6 =- 1) , f ( x ) = sin x- F ( x ) =- cos x, f ( x ) = cos x- F ( x ) = sin x, f ( x ) = 1 x 2 +1- F ( x ) = tan- 1 x, f ( x ) = 1 1- x 2- F ( x ) = sin- 1 x f ( x ) = e x- F ( x ) = e x , f ( x ) = 1 x- F ( x ) = ln | x | Examples Just like with differentiation, we can combine these basic antiderivatives into more complex ones. I will give a few examples first, and then explain some of the steps. Note that an antiderivative is easily checked: just differentiate it and see if you get the orginal function back. But I wont do that in this writeup. f ( x ) = 3 x 7- 5 sin( x )- F ( x ) = 3 1 8 x 8- 5 (- cos( x )) = 3 8 x 8 + 5 cos( x ) , f ( x ) = e 3 x + 2 1- x 2 + 1- F ( x ) = 1 3 e 3 x + 2 sec( x ) + x, f ( x ) = sin(2 x- 3)- F ( x ) =- 1 2 cos(2 x- 3) , f ( x ) = (3 x + 7) 5 + 1 x +1- F ( x ) = 1 6 (3 x + 7) 6 1 3 + ln | x + 1 | = 1 18 (3 x + 7) 6 + ln | x + 1 | 1 Compensating There is one informal trick that I used here several times; let me explain it for sin(2 x- 3). We can guess that an antiderivative should involve- cos(2 x- 3), but if we differentiate that, we would get 2 sin(2 x- 3) (with the 2 coming from the chain piece (2 x- 3) ), which is not quite what we want. However, we can fix it by compensating for the 2 by putting a 1 2 in front of- cos(2 x- 3). And indeed, then it works:- 1 2 cos(2 x- 3) =- 1 2 (- sin(2 x- 3) 2) = sin(2 x- 3) ....
View Full Document

Page1 / 8

105-1-AntiderivativesSubstitution - 1 Antiderivatives...

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