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
TECHNIQUES OF INTEGRATION
TECHNIQUES
SUBSTITUTION
f g(x) • g' (x) dx
f (x)
g ( x ) dx
u dv
choose function that easier to differentiate as u, remaining expression as dv
let u = g(x) then du=g'(x)dx substitute u and du into integral:
f (u) du
PARTIAL FRACTIONS
BY PARTS
find du from u (differentiate) and v from dv (integrate)
PROPER FRACTIONS decomposition of partial fractions
substitute into formula integrate with respect to u substitute back u (final answer in terms of x)
u dv = uv −
integrate
v du
integrate & simplify
PRIORITY OF CHOOSING U IN BY PARTS TECHNIQUES: USE ONE OF THESE TIPS: LoPET Log → Polynomial → Exponential → Trigonometry 0 . LATE 1 Log → Algebraic → Trigo → Exponential 5
LIATE Log → Inverse Trigo → Algebraic → Trigo → Exponential
LIPET Log → Inverse Trigo → Polynomial → Exponential → Trigo
IMPROPER FRACTIONS long division and decomposition
integrate