27 Chapter 8: TECHNIQUES OF INTEGRATION BASIC INTEGRATION FORMULAS TABLE: Differentiation Formula Integration Formula 1 1 ) ( 1 n c n x dx x x n n ) ( ' ) ( )] ( [ 1 x u x nu x u dx d n n 1 ) ( ' ) ( 1 n u du u dx x u x u n n n +C 1 n ) ( ' ) ( cos )) ( (sin x u x u x u dx d C x u udu dx x u x u ) ( sin cos ) ( ' ) ( cos ) ( ' ) ( sin )) ( (cos x u x u x u dx d C x u udu dx x u x u ) ( cos sin ) ( ' ) ( sin ) ( ' ) ( sec )) ( (tan 2 x u x u x u dx d C x u du u dx x u x u ) ( tan sec ) ( ' ) ( sec 2 2 ) ( ' ) ( csc )) ( (cot 2 x u x u x u dx d C x u du u dx x u x u ) ( cot csc ) ( ' ) ( csc 2 2 ) ( ' ) ( tan ) ( sec )) ( (sec x u x u x u x u dx d C x u du u u dx x u x u x u ) ( sec tan sec ) ( ' ) ( tan ) ( sec ) ( ' ) ( cot ) ( csc )) ( (csc x u x u x u x u dx d C x u du u u dx x u x u x u ) ( csc cot csc ) ( ' ) ( cot ) ( csc ) ( ' ) ( ) ( ) ( x u e e dx d x u x u c e dx x u e x u x u ) ( ) ( ) ( ' . ) ( ) ( ' )) ( (ln x u x u x u dx d c x u du x u dx x u x u ) ( ln ) ( 1 ) ( ) ( ' 0 ln ) ( ' ) ( ) ( ) ( a a x u a a dx d x u x u c a a dx x u a x u x u ) ( ) ( ln 1 ) ( ' . ) ( 1 ) ( ' )) ( (tan 2 1 x u x u x u dx d c a u a u a du dx u a u 1 2 2 2 2 tan 1 ' ) ( 1 ) ( ' )) ( (sin 2 1 x u x u x u dx d c a u u a du dx u a u 1 2 2 2 2 sin ' 1 ) ( ) ( ' )) ( (sec 2 1 x u u x u x u dx d c a u a a u u du dx a u u u 1 2 2 2 2 sec 1 '
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Chapter 8: TECHNIQUES OF INTEGRATION · Sec 8.1: Method of INTEGRATION BY PARTS From the product rule, we can obtain the following formula, which is very useful in integration: Start
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27
Chapter 8: TECHNIQUES OF INTEGRATION
BASIC INTEGRATION FORMULAS TABLE:
Differentiation Formula Integration Formula
1
1)(
1
ncn
xdxxx
nn
)(')()]([ 1 xuxnuxudx
d nn 1
)(')(1
n
uduudxxuxu
nnn
+C 1n
)(')(cos))((sin xuxuxudx
d Cxuududxxuxu )(sincos)(')(cos
)(')(sin))((cos xuxuxudx
d Cxuududxxuxu )(cossin)(')(sin
)(')(sec))((tan 2 xuxuxudx
d Cxuduudxxuxu )(tansec)(')(sec 22
)(')(csc))((cot 2 xuxuxudx
d Cxuduudxxuxu )(cotcsc)(')(csc 22
)(')(tan)(sec))((sec xuxuxuxudx
d Cxuduuudxxuxuxu )(sectansec)(')(tan)(sec
)(')(cot)(csc))((csc xuxuxuxudx
d Cxuduuudxxuxuxu )(csccotcsc)(')(cot)(csc
)(')( )()( xueedx
d xuxu cedxxue xuxu )()( )('.
)(
)('))((ln
xu
xuxu
dx
d cxudu
xudx
xu
xu)(ln
)(
1
)(
)('
0ln)(')( )()( aaxuaadx
d xuxu ca
adxxua xuxu )()(
ln
1)('.
)(1
)('))((tan
2
1
xu
xuxu
dx
d
ca
u
aua
dudx
ua
u 1
2222tan
1'
)(1
)('))((sin
2
1
xu
xuxu
dx
d
ca
u
ua
dudx
ua
u
1
2222sin
'
1)(
)('))((sec
2
1
xuu
xuxu
dx
d c
a
u
aauu
dudx
auu
u
1
2222sec
1'
28
Sec 8.1: Method of INTEGRATION BY PARTS
From the product rule, we can obtain the following formula, which is very useful in integration:
Start with vdx
du
dx
dvuxvxu
dx
d))'()(( .
Integrate both sides to get vduudvuv
Rearrange to obtain the integration by parts formula
To integrate by parts, strategically choose u, dv and then apply the formula.
Key Concept
Choose u, dv in such a way that:
o u is easy to differentiate.
o dv is easy to integrate.
o
Sometimes it is necessary to integrate by parts more than once
ILATE are supposed to suggest the order in which you are to choose the “u:
for choosing which of two functions is to be u and which is to be dv is to choose u by whichever
function comes first in the following list:
I: inverse trigonometric functions: arctan x , arcsec x, etc.
L: the logarithmic function: ln x
A: algebraic functions: x2,3x50, etc.
T: trigonometric functions: sin x, tan x, etc.
E: exponential functions: ex, 13x, etc.
Integration by parts ``works'' on definite integrals as