Chapter 4. Section 7
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C. Bellomo, revised 17-Sep-08
Section 4.7 – Antiderivatives
A Recap of Some Derivatives:
• Let’s recall some derivatives we have seen so far…
44
22
sin cos
1
ln
4
cos(2)(41)sin(2)
xx
dxx
dx
dx
dx x
dee
dx
d
x
xx xx
dx
=
=
=
+=− + +
• If we wanted to ‘undo’ the derivative, we could say
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22
Antiderivative[cos ] sin
1
Antiderivative ln
Antiderivative[4 ]
Antiderivative[ (4 1)sin(2 )] cos(2 )
xx
xx
x
x
ee
x
xx xx
=
⎡⎤
=
⎢⎥
⎣⎦
=
−+ += +
• In this section we will not focus much on finding this antiderivative, we will just be identifying the
relationship between them.
The Arbitrary Constant:
• Look at the functions below with their derivatives, what do you notice?
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44
(sin 17) cos
(sin 15) cos
1
(12 ln )
1
(ln)
(7)4
(8)4
xx
xx
d
x
x
dx
d
x
x
dx
dx
dx x
dex
dx x
dee
dx
dee
dx
+=
−=
+=
+=
−=
+=
• Adding (or subtracting) a constant to a function only shifts it up or down. It does not affect the
derivative (or shape) of that function.
