4.9 Antiderivatives
Thurs Jan 7
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
2
d 1 3
( x )= x
dx 3
( x + 5) = x
• When integrating, we must take into
consideration the constant that went
away
d
dx
1
3
3
2
Indefinite Integral
• Let F(x) be any antiderivative of f. The
indefinite integral of f(x) (with respect to
x) is defined by
ò f (x)dx = F(x) + C
where C is an arbitrary constant
Examples
• Examples 1.2 and 1.3
ò 3x dx
2
ò t dt
5
The Power Rule
• For any rational power
ò
r ¹ -1
r+1
x
x dx =
+C
r +1
r
• 1) Exponent goes up by 1
• 2) Divide by new exponent
Examples
• Examples 1.4, 1.5, and 1.6
òx
17
dx
ò
1
dx
3
x
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 + b ò g(x)dx
You try
ò 3x - 2x +1 dx
2
ò
1
x3
+ x dx
Trigonometric Integrals
• These are the trig integrals we will work
with:
ò sin x dx = -cos x + C
ò cos x dx = sin x + C
ò sec x dx = tan x + C
ò csc x dx = -cot x + C
2
2
ò sec x tan x dx = sec x + C
ò csc x cot x dx = -csc x + C
Exponential and Natural Log
Integrals
• You need to know these 2:
ò e dx = e
x
ò
x
+C
1
dx = ln x + C
x
Example
• Ex 1.8
ò (3e
x
- 2sec x) dx
2
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
Form f(ax)
• If ò f (x) dx = F(x) + C , then for any
constant a ¹ 0,
ò
1
f (ax) dx = F(ax) + C
a
• 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’
Examples
• Ex 1.9
ò sin 3x dx
ò 5e
4x
dx
ò 8sec
2
5x dx
Extra Do Now if needed –
ignore this
• Do Now
• Integrate
• 1) ò sin3x + 3x 3 dx
• 2) ò cos x - e 2x 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) = 3x -1,
2
f (0) = 2 f ¢(x) = 2cos x
f (0) =1
You try
Find the original function
f ¢(x) = 3x + x + 3,
2
f (0) = 5
dy
7
= 4x , y(0) = 4
dx
Finding f(x) from f’’(x)
• When given a 2nd 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)
2
¢
f (x) = x - sin x,
f (0) = 3
• HW: p.280 #1-71 odds
4-9 Anti-derivatives
Fri Jan 8
• Do Now
• Integrate and find C
• 1) ò sin3x + 3x 3 dx, f (0) =1
• 2)
ò cos x - e
2x
dx,
f (0) = 2
HW Review
Closure
• Journal Entry: What is integration? How
are integrals and derivatives related?
Why do we include +C?
• HW:
• CH 4 AP MC and FRQ due Mon/Tues