integration by guessing by fred halpern [email protected] royalpathtomath.org ---- web site
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Integration by Guessing By Fred Halpern
RoyalPathToMath.Org ---- Web Site
Why Guess
• Eliminates boring , fussy calcs• Sub and by parts have a hidden guess.
Disclaimer. - Does not replace the usual Techniques - But uses them more Effectively
Lucky Guess
Problem Find
Sub We See SOL.
2dx
1
x
x
2 2
2We Guess 1 . 1
1
xx D x
x
1
2u
2
2 dx 1 C .
1
xx
x
Solution
21 .u x 1
2u u
Desperate Guess
Problem 4.1.
Exact hit Problem Solved !
Find (x lnx + x )dx x xLearn more about x x 1x x lnx + x x .x x xD
(x lnx + x )dx xx x x C
Fundamental Theorem of the Calculus
Method of Guessing • Guess• Differentiate to check • Adjust
Theorem
If D F(x) = f(x) , then
f x dx F x C
Off by a Factor
Problem Find
D =
2x 1 5 dxx3
2 2(1 5 )x
12 215 (1 5 )x x
Solution
32 2
2 (1 5 )x 1 5 dx
15
xx C
We See SOL
Guess
3
2u1
2u
32 2(1 5 )x
Easy by Guessing
Problem 3.1.
Find
2 23 3cos(2 5 )dxx xxe e2
2 2 2
3
3 3 3
Guess sin(2 5 ).
sin(2 5 30 x (2 5 ).) cos
x
x x x
e
D e e e
Adjust Divide by 302
32 23 3 sin(2 5 )
cos(2 5 )dx30
x
x x exe e C
Eliminate Extra Summand
Theorem : If D F(x) = f(x) + R(x), then
Theorem (Linearity)
If DF(x) = K f(x) , then
f x dx F x x dx.R
( )f(x)dx
F xC
K
Inverse Functions, I(x)
Find , I(x) Inverse function :x (x) n I dxAlways guess
1x (x) ( (x), x )
1
nnI
u I dvn
2 2
Problem: Find ln dx
Guess ln . ln ln .2 2 2
Solution. ln dx ln dx2
x x
x x xx D x x x
xx x x x C
By Parts – Inverse function
Problem. Find arctan( )dxx
2
2
2
arctan( )
arctan( ) arctan( )
Solution. arctan( ) dx = arctan( ) -1
ln(1 )arctan( ) -
2
Guess .
1
x x
x x x x
xx x x dx
x
xx x C
xD
x
Integration by parts
Problem. Find cos( )dxx x
Solution. cos( )dx sin sin dxx x x x x
sin x cos x sin x . Dx x
( , cos( ))
cos( )dx = sin x, Guess x sin x .
u x dv x
x
Arc Trig Functions
2 2 2
arcsin( )1dx . Guess arcsin( )
bx ba x
b aa b x
2 2 2
arctan( )1dx . Guess arctan( ).
bx ba x
a b x ab a
2 2 2
arcsec( )1dx
bx
aax b x a
Guess arcsec( )bx
a
Kitchen Sink Problem
Find
Setting u = arcsin(x). We get arccos(x) = .
Also x= sin u, = cos u , dx=cos u du.
Substitution yields
Solution 4.6.
2( 1 ) arcsin x arccos x dxp q m nx x
21 x
sin cos u ( ) cos du2
p q m nu u u u
2( 1 ) arcsin x arccos x dxp q m nx x
2u