4.9 Antiderivatives
Wed Feb 4
Do Now
Find the derivative of each function
1)
f (x) = x
2)
2
f (x) = cos 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)
• Antiderivatives are also known as
integrals
Integrals + C
• When differentiating, constants go away
d
dx
( x )= x
d
dx
( x + 5) = x
1
3
1
3
3
3
2
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
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
EX
2
ò 3x - 2x +1 dx
ò
1
x3
+ x dx
Closure
• Hand in: Integrate the following function
f (x) = ò 2x + 3x - 5 dx
3
• HW: p. 280 #1-2 11-23 odds
4-9 Integrals of Trig, e, lnx
Thurs Feb 5
• Do Now
• Integrate the following:
• 1) 9x 80 dx
ò
2
• 2)
ò x dx
3
HW Review: p.280 #1, 2, 1123 odds
•
•
•
•
•
•
•
•
1) 6x + C
23) t
5x 2/5
2)
2 +C
2
11) 4x - 9x + C
11t 5/11
13) 5 + C
6
5
2
15) 3t - 2t -14t + C
1/5
3z 5/3
4z 9/4
17) 5z - 5 + 9 + C
2/3
3x
19)
2 +C
-2
21)
3
-18t + C
2 5/2
5
+ t + t
1 2
2
2 3/2
3
+ t +C
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
Examples
• Ex 1.7
ò (3cos x + 4x ) dx
8
Exponential and Natural Log
Integrals
• You need to know these 3:
ò e dx = e
x
x
+C
ò e dx = -e + C
1
ò x dx = ln x + C
-x
-x
Example
• Ex 1.8
ò (3e
x
- 2sec x) dx
2
You try
• Integrate the following:
• 1) 2sin x - 2x +1 dx
ò
• 2)
• 3)
ò 5x
ò 3e
x
-1
- x dx
- 2x
1/ 2
dx
Closure
• Hand in: Integrate the following
ò (cos x + 4 / x) dx
• HW: p. 280 #3-9 odds 26-29 all 36
4-9 Integrals of the form f(ax)
Fri Feb 6
• Do Now
• Evaluate the following integrals
ò 4x
8
dx
ò (sin x + cos x) dx
2
ò 3e - x dx
x
HW Review p.280 3-9 26-29
36
5
3
2
• 3) 5 x -8x +12 ln x + C
•
•
•
•
•
•
•
5) 2sinx + 9cosx + C 36) 4lnx – e^x + C
7) 12e x + 5x -1 + C
9) a-ii b-iii c-i d-iv
26) - 13 cos x - 14 sin x + C
27) 12sec x + C
28) 12 q 2 + tan x + C
29) –csc t + C
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
Examples
• Ex 1.9
ò sin 3x dx
ò 5e
4x
dx
ò 8sec
2
5x dx
You Try
• Evaluate the integrals
ò 4 cos5x dx
ò 4 sin 3x -1 dx
ò 3sec2x tan2x dx
Closure
• Hand in: Integrate the following
3x
ò sin2x + e
dx
• HW: p.281 #31-39 odds, 30 38
• Quiz Next Thurs
4-9 Finding original functions
through integrating
Mon Feb 9
• Do Now
• Integrate
• 1) ò sin3x + 3x 3 dx
• 2) ò cos x - e 2x dx
HW Review p.281 #30-39
•
•
•
•
•
•
•
1
30)
7 cos(7x - 5)+C
1
31) 3 tan(7 - 3x)+C
33) 25
3 tan(3z +1)+C
q )+ C
1
sin3
q
2tan(
35) 3
4
3 5x
37) 5 e + C
3t-4
1
38) 3 e
+C
39) 4x 2 + 2e5-2 x + C
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
Closure
• Hand in: Find the original function of
2
¢
f (x) = x - sin x, f (0) = 3
• HW: p.281-282 #47-61 odds
• 4.9 Quiz Thurs Feb 12
4-9 Working from the
derivative
Tues Feb 10
• 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
nd
2
HW Review p.281-2 #47-61
•
•
•
•
•
•
•
•
47) y = x + 4
3
49) y = t + 3t - 2
3/2
2
51) y = 3 t + 13
4
1
53) y = 12 (3x + 2) - 13
55) y =1- cos x
1
y
=
57)
5 sin5x + 3
x
2
59) y = e - e
12-3t
61) y = -3e
+10
1
4
2
4
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
Example 1
• If a space shuttle’s downward
acceleration is given by y’’(t) = -32
ft/s^2, find the position function y(t).
Assume that the shuttle’s initial velocity
is y’(0) = -100 ft/s, and that its initial
position is y(0) = 100,000 ft.
Ex 2
• A car traveling with velocity 24m/s
begins to slow down at time t = 0 with a
constant deceleration of a = -6 m/s^2.
When t = 0, the car has not moved. Find
the velocity and position at time t.
Closure
• Hand in: Determine the position function
if the acceleration function is a(t) = 12,
the initial velocity is v(0) = 2, and the
initial position is s(0) = 3
• HW: p.282 #63-69 odds
• 4.9 Quiz Thurs Feb 12
4.9 Review
Wed Feb 11
• Do Now
• If a ball is thrown up into the air and
begins to fall, it has an acceleration
function of a(t) = -32 ft/s^2. Find the
position function if the initial velocity is
v(0) = 0, and its initial position is
s(0) = 20 ft
HW Review p.282 #63-69
• 63) f
'(x) = 6x +1
2
f (x) = 2x + x + 2
2
1 4
f
'(x)
=
x
x
+ x +1
• 65)
4
5
1
1 3
1 2
f (x) = 20 x - 3 x + 2 x + x
• 67)
f '(t) = -2t -1/2 + 2
f (t) = -4t1/2 + 2t + 4
1 2
f
'(t)
=
• 69)
2 t - sin t + 2
f (t) = 16 t 3 + cost + 2t - 3
3
Integral Quiz Review
• What to know:
–
–
–
–
–
–
–
Power Rule
Trig Rules (sinx, cosx, sec^2 x)
The two exponential rules
Ln x
Sums and differences of integrals
Integral of f(ax)
Solving for C
• 2nd deriv / Acceleration may be included in this section
Review
• Worksheet p.332 #1-24 27 29-32
#55-60 65-68
+C
Closure
• Journal Entry: What is integration? How
are integrals and derivatives related?
• HW: Finish worksheet
• Quiz Thurs Feb 12
4.9 Review
Tues Feb 11
• Do Now
• Given f ’’(x) = -32, f ‘ (0) = 2, and f(1) =
5, find f(x)
•
•
•
HW Review: p.332
#5,7,10,11,12
5) 3x + C
5
7) 3x - 3x + C
5
2
10) 2x 2 - 4 x 3 / 2 + C
3
• 11)
• 12)
5
5
2
x -3
3
-1
3x + + C
1/ 2
-2x + 2x + C
HW Review p.332 #15 16 19
21 23
•
•
•
•
•
15) -2cosx + sinx + C
16) 3sinx + cosx + C
19) 5tanx + C
x
21) 3e - 2x + C
23) 3sinx - ln|x| + C
HW Review p.332 #27 29 31
34 39
-x
5x
• 27) 2 + 3e + C
5
• 29) - 2 cos2x + C
• 31) e3 - x2 + C
2
3x
2
• 34) tan3x + C
2 5/2
16
x
• 39) 5
5
x
5/4
+C
HW Review p.333 #55-60
•
•
•
•
•
•
55) f (x) = x - x + 2
56) f (x) = 4sin x2 + 3
x
x
57) f (x) = 3e + + 1
2
3
5
f
(x)
=
cos2x
+
58)
2
2
2
59) f (x) = 6x + 2x + 3
1 3
60) f (x) = 3 x - 3x + 2
4
3
3
HW Review p.333 #65-68
s(t) = 3t - 6t + 3
2
• 65)
•
• 66)
s(t) = -3e - 2t + 3
• 67)
s(t) = -3sint + t + 3t + 4
• 68)
-t
1 2
2
s(t) = t + t + 4t
1 4
12
1 2
2