short notes module 4 integ

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SHORT NOTES MODULE 4: INTEGRATION INTEGRATION

 f(x) dx = F(x) + c

DEFINITE INTEGRAL

b

 f(x) dx = F(b) − F(a)

INDEFINITE INTEGRAL

a

BASIC RULES OF INTEGRATION:

 

k dx = kx + c

( ax + b )n dx =

kf(x) dx = k f(x) dx

( ax + b )n+1 + c (n + 1)( a )

kxn dx =

kxn+1 +c n +1

; n  −1

 f(x)  g(x) dx =  f(x) dx   g(x) dx

; n  −1

INTEGRATIONS WITH RESULT IN SPECIFIC FUNCTIONS:

EXPONENTIAL

LOGARITHM

TRIGONOMETRY

INVERSE TRIGONOMETRY

 

k ax +b +c aln k k is constant

k ax +b dx =

eax +b dx =

ln ax + b 1 dx = +c ax + b a

 sincos(axax+ b+) bdx

( ax + b ) dx  cos sin ax + b

=−

 =

(

) +c

a 1 1 − ( ax ) sin−1 ( ax )

2

a

1 dx = ln x + c x

(

=

) +c

a

dx

+c

=

−1 1 − ( ax ) cos−1 ( ax ) 2

a

e

eax +b +c a

dx = e x + c

x

f ' (x)

 f ( x ) dx = ln f ( x ) + c ( ax + b ) dx  sec tan ax + b 2

(

=

) +c

a

dx

 1 + (ax )

+c

=

1

2

dx

tan−1 ( ax ) a

+c


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