antiderivatives an antiderivative of f(x) is any function f(x) such that f’(x) = f(x)
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Theorem
• If F(x) is an antiderivative of f(x)
then F(x) + C is an antiderivative of f(x) for any constant C
The Fundamental Theorem of Calculus, Part 1
If f is continuous on , then the function ,a b
x
aF x f t dt
has a derivative at every point in , and ,a b
x
a
dF df t dt f x
dx dx
a
xdf t dt
xf x
d
2. Derivative matches upper limit of integration.
First Fundamental Theorem:
1. Derivative of an integral.
a
xdf t dt f x
dx
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
First Fundamental Theorem:
x
a
df t dt f x
dx
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
New variable.
First Fundamental Theorem:
cos xd
t dtdx cos x 1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
The long way:First Fundamental Theorem:
20
1
1+t
xddt
dx 2
1
1 x
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
2
0cos
xdt dt
dx
2 2cosd
x xdx
2cos 2x x
22 cosx x
The upper limit of integration does not match the derivative, but we could use the chain rule.
53 sin
x
dt t dt
dxThe lower limit of integration is not a constant, but the upper limit is.
53 sin xdt t dt
dx
3 sinx x
We can change the sign of the integral and reverse the limits.
2
2
1
2
x
tx
ddt
dx eNeither limit of integration is a constant.
2 0
0 2
1 1
2 2
x
t tx
ddt dt
dx e e
It does not matter what constant we use!
2 2
0 0
1 1
2 2
x x
t t
ddt dt
dx e e
2 2
1 12 2
22xx
xee
(Limits are reversed.)
(Chain rule is used.)2 2
2 2
22xx
x
ee
We split the integral into two parts.
The Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of , and if
F is any antiderivative of f on , then
,a b
b
af x dx F b F a
,a b
(Also called the Integral Evaluation Theorem)
To evaluate an integral, take the anti-derivatives and subtract.
Antiderivatives
• Antiderivatives are also called indefinite integrals
• They are sometimes written
• Note that there are no limits on the integral
• Do not confuse with definite integrals!
F x f x dx
Common Antiderivatives
2
32
1
1)
2)2
3)3
4)1
15) ln
nn
dx x C
xxdx C
xx dx C
xx dx C
n
dx x Cx
2
2
6) sin cos
7) cos sin
8) sec tan
9) csc cot
10) sec tan sec
11) csc cot csc
xdx x C
xdx x C
xdx x C
xdx x C
x xdx x C
x xdx x C
Rewrite then evaluate the integral using FTC2
4
0
2
41
22
0
2 2
31
2 2
1
1)
32)
3) 1
44)
15)
xdx
dtt
y dy
udu
u
xdx
x
Evaluate the integral involving trigonometric functions using FTC22
32
6
2
3
24
20
1) cos
2) csc
3) csc cot
1 cos4)
cos
d
d
x xdx
udu
u
Area using Integrals
• Find the zeros of the function over the interval [a,b]
• integrate over each subinterval
• add the absolute value of the integrals
Using the GC to find the integral
• hit MATH then 9
• fnInt( will come up on the screen
• type in the function, comma, x, comma, -a, comma, b) then hit ENTER
• Ex: 2
1
sin ( sin( ), , 1,2) 2.043x xdx fnINT x x x
Area using GC
• To find the area under the curve f(x) from [a,b] type fnInt(abs(f(x)),x,a,b)
• Example: Find the area under the curve
y = xcos2x on [-3, 3]
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