The Integral Notation ∫

u cscu secu du = −cscu + C u secu tanu du = secu + C u csc2 u du = −cotu + C u. 1 − u2 du = arcsinu + C. −u. 1 − u2 du = arccosu + C u. 1 + u2 du = arctanu + C.
Télécharger le PDF
5MB taille 3 téléchargements 151 vues
commentaire
Report
The Integral Notation ∫

!

𝑓(𝑥!∗ ) 𝛥𝑥

lim

!→∞

!!!

!

≡

𝑓(𝑥) 𝑑𝑥 !



Definite Integral Properties !

!

𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎

𝑐 𝑑𝑥 = 𝑐 𝑏 − 𝑎

!

!

!

!

𝑓 𝑥 𝑑𝑥 = 0

!

𝑐𝑓 𝑥 𝑑𝑥 = 𝑐

!

𝑓 𝑥 𝑑𝑥

!

!

!

!

𝑓 𝑥 𝑑𝑥 = 0

!

𝑓 𝑥 ± 𝑔 𝑥 𝑑𝑥 =

!!

!

⇔ 𝑓 −𝑥 = −𝑓 𝑥 ! !!

!

𝑓 𝑥

!

𝑔 𝑥 𝑑𝑥 !

!

𝑓 𝑥 𝑑𝑥 + !

𝑓 𝑥 𝑑𝑥 𝒌

even

NOTE: 𝑓 𝑥 ⋅ 𝑔 𝑥 𝑑𝑥 ≠

𝒌

𝑓 𝑥 𝑑𝑥 =

!

⇔ 𝑓 −𝑥 = 𝑓 𝑥

𝑓 𝑥 𝑑𝑥 ± !

odd

!

𝑓 𝑥 𝑑𝑥 = 2

!

𝑔 𝑥 𝑑𝑥 ⋅

!

𝑓 𝑥 𝑑𝑥

!

𝑓 𝑥 𝑑𝑥 = − !

𝑓 𝑥 𝑑𝑥 !

Fundamental Theorems Let 𝑓 𝑥 = 𝑢 and 𝑔 𝑥 = 𝑣 for the following: !

𝑖)

𝑦=

𝑓 𝑡 𝑑𝑡 ⇒

𝑦 ! = 𝑓 𝑣 ∙ 𝑣 ! − 𝑓 𝑢 ∙ 𝑢′

!



!

𝑦=

𝑦 ! = 𝑓 𝑣 ∙ 𝑣 ! − 𝑓 𝑎 ∙ 𝑎! = 𝑓 𝑣 ∙ 𝑣 ! − 0 = 𝑓 𝑣 ∙ 𝑣 !

𝑓 𝑡 𝑑𝑡 ⇒ !



!

𝑦=



𝑦 ! = 𝑓 𝑏 ∙ 𝑏 ! − 𝑓 𝑢 ∙ 𝑢! = 0 − 𝑓 𝑢 ∙ 𝑢! = −𝑓 𝑢 ∙ 𝑢′

𝑓 𝑡 𝑑𝑡 ⇒ !



Limit Definition of a Definite Integral !

𝑖𝑖)

𝑓(𝑥!∗ ) 𝛥𝑥 =

lim

!→∞

!!!

!

𝑓(𝑥) 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎 !

𝑏−𝑎 𝛥𝑥 = , 𝑛



Differential Equation (1st order) 𝑦! = 𝑓! 𝑥 ⇒

𝑥! = 𝑎 + 𝑖 ∙ 𝛥𝑥

𝑑𝑦 = 𝑓 ! 𝑥 ⇒ 𝑑𝑦 = 𝑓 ! 𝑥 𝑑𝑥 ⇒ 𝑑𝑥

𝑑𝑦 =

𝑓 ! 𝑥 𝑑𝑥

⇒ 𝑦 + 𝑐! = 𝑓 𝑥 + 𝑐! ⇒ 𝑦 = 𝑓 𝑥 + 𝑐! − 𝑐! = 𝑓 𝑥 + 𝑐! ≡ 𝑓 𝑥 + 𝐶 Common Integrals 𝑑𝑥 = 𝑥 + 𝐶

𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶

1 𝑥 ! 𝑑𝑥 = 𝑥 ! + 𝐶 3

𝑥 ! 𝑑𝑥 =

1 𝑥 𝑑𝑥 = 𝑥 ! + 𝐶 2

1 𝑥 !!! + 𝐶 𝑛+1

1 𝑑𝑥 = ln |𝑥| + 𝐶 𝑥

⇔ 𝑛 ≠ −1 1 !" 𝑒 + 𝐶 𝑎

𝑒 ! 𝑑𝑥 = 𝑒 ! + 𝐶

𝑒 !" 𝑑𝑥 =

1 𝑑𝑥 = ln 𝑥 + 1 + 𝐶 𝑥+1

1 1 𝑑𝑥 = ln 𝑎𝑥 + 𝑏 + 𝐶 𝑎𝑥 + 𝑏 𝑎

𝑒 ! 𝑢′ 𝑑𝑢 = 𝑒 ! + 𝐶

𝑢! 𝑑𝑢 = ln 𝑢 + 𝐶 𝑢

1 !"!! 𝑒 + 𝐶 𝑎

𝑓 𝑢 𝑢′ 𝑑𝑢 = 𝐹 𝑢 + 𝐶 !

𝑓 𝑥 =𝐹 𝑏 −𝐹 𝑎 !

𝑢! cos 𝑢 𝑑𝑢 = sin 𝑢 + 𝐶

𝑢! sin 𝑢 𝑑𝑢 = − cos 𝑢 + 𝐶

𝑢! sec ! 𝑢 𝑑𝑢 = tan 𝑢 + 𝐶

𝑢! csc 𝑢 sec 𝑢 𝑑𝑢 = − csc 𝑢 + 𝐶

𝑢! sec 𝑢 tan 𝑢 𝑑𝑢 = sec 𝑢 + 𝐶

𝑢! csc ! 𝑢 𝑑𝑢 = − cot 𝑢 + 𝐶

𝑢! 1 − 𝑢!

𝑑𝑢 = arcsin 𝑢 + 𝐶

−𝑢! 1 − 𝑢!

𝑑𝑢 = arccos 𝑢 + 𝐶



𝑒 !"!! 𝑑𝑥 =

𝑢! 𝑑𝑢 = arctan 𝑢 + 𝐶 1 + 𝑢!