fundamental theorem of calculus chain rule

fundamental theorem of calculus chain rule

INTEGRATION FORMULAS:
! f (x) dx = F(x) + C, where F ' (x) = f (x)
d
dx
!
x
a
f (t) dt = f (x)
Remember the Chain Rule!!!:
"
b
a
[First Fundamental Theorem]
d u
f (t) dt = f (u) " Du
dx ! a
f (x) dx = F(b) ! F(a), where F ' (x) = f (x)
[Second Fundamental Theorem]
u n+1
! u du = n + 1 + C, n " #1
!
u
u
! e du = e + C
u
! a du =
" sinu du = ! cosu + C
! cosu du = sinu + C
! sec u du = tan u + C
! csc u du = " cot u + C
! sec u • tan u du = secu + C
" csc u • cot u du = ! cscu + C
! secu du = ln
sec u + tan u + C
! tan u du = ln
sec u + C
" cscu du = ln
csc u ! cot u + C
! cot u du = ln
sinu + C
n
2
1
1
! sin u du = 2 u " 4 sin 2u + C
2
"
"u
du
#u
= sin !1 $ %& + C
2
a
a !u
2
du
1
!1 u
=
•
sec
+C
a
u2 ! a 2 a
" u dv = uv ! " v du
du
= ln u + C
u
au
+ C, ( a > 0, a " 1)
ln a
2
1
1
! cos u du = 2 u + 4 sin 2u + C
2
!a
2
"u
du
1
#u
• tan "1 $ %& + C
2 =
+u
a
a
2
du
1 # u ! a&
ln %
(+C
2 =
!a
2a $ u + a '