Properties of Sums and Integrals - SLC Home Page

MATHEMATICS 201-203-RE
Integral Calculus
Martin Huard
Winter 2009
Properties of Sums and Integrals
Properties of Finite Sums
1.
n
∑
n
∑
ca = c a
i
i= m
i=
m
n n n
∑ ∑ ∑
2. ( )
i i i i
i= m i= m i=
m
i
a ± b = a ± b
Special Sums
1.
3.
5.
n
∑ 1 = n
2.
i=
1
i=
1
( + 1)
n
n n
∑ i =
4.
2
( + 1) ⎤
2
⎡ n n
∑ n
3
i = ⎢ ⎥
i=
1 2
6.
⎣
⎦
n
∑
i=
1
n
∑
i=
1
n
∑
i=
1
c = cn
i
i
2
4
n n
=
( + 1)( 2n
+ 1)
6
( + 1)( 2 + 1)( 3 2 + 3 −1)
n n n n n
=
30
The Definite Integral
b
n
⎛
* ⎞
∫ f ( x) dx = lim⎜∑
f ( xi
) ∆x
⎟ where
n→∞ ⎝ i=
1 ⎠
a
b − a
∆ x =
n
*
and x [ x x ]
∈
−1,
with x i
= a + i∆
x
i i i
The Fundamental Theorem of Calculus
Suppose f is continuous on [ a,
b ] .
x
1. If g ( x) = ∫ f ( t)
dt , then g ( x) f ( x)
a
b
2. ( ) = ( ) − ( )
′ = .
∫ f x dx F b F a , where F is any antiderivative of f, that is, F′ = f .
a

Properties of the integral
∫ ∫
1. ( ) = ( )
cf x dx c f x dx
∫ ∫ ∫
2. ⎡ ( ) ± ( ) ⎤ = ( ) ± ( )
⎣ f x g x ⎦ dx f x dx g x dx
b
a
3. ( ) = − ( )
∫a
∫ b
f x dx f x dx
c b b
4. ( ) + ( ) = ( )
∫ ∫ ∫
f x dx f x dx f x dx
a c a
b
5. If f ( x) ≥ 0 for a ≤ x ≤ b , then ( ) 0
∫ f x dx ≥ .
a
b
b
6. If f ( x) ≥ g ( x)
for a ≤ x ≤ b , then ( ) ≥ ( )
∫ f x dx g x dx
a ∫ .
a
b
7. If m ≤ f ( x)
≤ M for a ≤ x ≤ b , then m( b − a) ≤ f ( x) dx ≤ M ( b − a)
8. udv = uv − vdu
∫ .
a
b
b b
∫ ∫ ∫ udv = uv ] − vdu
a
a ∫ (Integration by parts)
a
Table of Indefinite Integrals
1.
n+
1
n x
∫ x dx = + C n ≠ − 1
2.
n + 1
∫
1
dx = ln x + C
x
3.
e x
dx e x
∫ = + C
4.
∫
x
x a
a dx = + C
ln a
5. ∫ sin xdx = − cos x + C
6. ∫ cos xdx = sin x + C
7.
sec 2
∫ xdx = tan x + C
8.
∫
csc
2
xdx = − cot x + C
9. ∫ sec x tan xdx = sec x + C
10. ∫ csc x cot xdx = − csc x + C
11. ∫ sec xdx = ln sec x + tan x + C 12. ∫ csc xdx = ln csc x − cot x + C
13. ∫ tan xdx = ln sec x + C
14. ∫ cot xdx = ln sin x + C
1
15. ∫ dx = arctan x + C
16.
2
1+
x
∫
1
1−
x
2
dx = arcsin x + C