4.9 Antiderivatives
Tues Dec 1
Do Now
If f ’(x) = x^2, find f(x)

Antiderivatives
• Antiderivative - the original function in a derivative problem (backwards)
• F(x) is called an antiderivative of f(x) if
F’(x) = f(x)
• g(x) is an antiderivative of f(x) if g’(x)
= f(x)
• Antiderivatives are also known as integrals

Integrals + C
• When differentiating, constants go away d dx
( 1
3 x
3
)
= x
2 d dx
( 1
3 x 3 +
5)
= x 2
• When integrating, we must take into consideration the constant that went away

Indefinite Integral
• Let F(x) be any antiderivative of f. The indefinite integral of f(x) (with respect to x) is defined by
f ( x ) dx
=
F ( x )
+
C where C is an arbitrary constant

Examples
• Examples 1.2 and 1.3
ò
3 x
2 dx
ò t
5 dt

The Power Rule
• For any rational power
x r dx
= x r
+
1 r
+
1 r
¹ -
1
+
C
• 1) Exponent goes up by 1
• 2) Divide by new exponent

Examples
• Examples 1.4, 1.5, and 1.6
ò x
17 dx
x
1
3 dx
ò xdx

The integral of a Sum
• You can break up an integrals into the sum of its parts and bring out any constants
ò
[ af ( x )
+ bg ( x )] dx
= a
ò f ( x ) dx
+
ò
( x ) dx

You try
ò
3 x
2 -
2 x
+
1 dx
ò
1 x
3
+ x dx

Trigonometric Integrals
• These are the trig integrals we will work with:
ò sin x dx
= cos x
+
C
ò
ò
ò cos x dx sec csc
2 x dx
= sin x
+
C
= tan x
+
C
ò
ò
2 x dx
= cot x
+
C sec x tan x dx csc x cot x dx
= sec x
+
C
= csc x
+
C

Exponential and Natural Log
Integrals
• You need to know these 2:
ò e x dx
= e x +
C
ò 1
= ln x
+
C x dx

Example
• Ex 1.8
ò
(3 e x -
2sec
2 x ) dx

Integrals of the form f(ax)
• We have now seen the basic integrals and rules we ’ ve been working with
• What if there ’ s more than just an x inside the function? Like sin 2x?

Integrals of Functions of the
ò
Form f(ax) f ( x ) dx
=
F ( x )
+
C a
¹
0
f ( ax ) dx
=
1 a
F ( ax )
+
C
• Step 1: Integrate using any rule
• Step 2: Divide by a

Note
• While this works for “basic” chain rule functions, it does not work for anything more than a linear ‘inside’

ò
• Ex 1.9
sin 3 x dx
Examples
ò
5 e
4 x dx
ò
8sec
2
5 x dx

Extra Do Now if needed – ignore this
• Do Now
• Integrate
• 1)
ò sin 3 x
+
3 x
3 dx
• 2)
ò cos x
e
2 x dx

Revisiting the + C
• Recall that every time we integrate a function, we need to include + C
• Why?

Solving for C
• We can solve for C if we are given an initial value.
• Step 1: Integrate with a + C
• Step 2: Substitute the initial x,y values
• Step 3: Solve for C
• Step 4: Substitute for C in answer

Examples f
¢
( x )
=
3 x
2 -
1, f (0)
=
2 f
¢
( x )
=
2cos x f (0)
=
1

You try
Find the original function f
¢
( x )
=
3 x
2 + x
+
3, f (0)
=
5 dy
=
4 x
7
, y (0)
=
4 dx

Finding f(x) from f’’(x)
• When given a 2 nd derivative, use both initial values to find C each time you integrate
• EX: f’’(x) = x^3 – 2x, f’(1) = 0, f(0) = 0

Acceleration, Velocity, and
Position
• Recall: How are acceleration, velocity and position related to each other?

Integrals and Acceleration
• We integrate the acceleration function once to get the velocity function
– Twice to get the position function.
• Initial values are necessary in these types of problems

Closure
• Find the original function f(x) f
¢
( x )
= x
2 sin x , f (0)
=
3
• HW: p.280 #3 9 17 25 33 37 47 51 59
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