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Antiderivatives. Zerick Lucernas Property
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ANTIDERIVATIVES
By:Zerick B. Lucernas
IV-Einstein
Introduction
In each case, the problem is to find a function F whose derivative is a known function f.
If such a function F exists, it is called an antiderivative of f.
Definition
A function F is called an antiderivative of f on an interval I if F’(x) = f (x) for all x in I.
For instance, let f (x) = x2. It is not difficult to discover an antiderivative of f if we keep the Power Rule in mind.
In fact, if F(x) = ⅓ x3, then F’(x) = x2 = f (x).
However, the function G(x) = ⅓ x3 + 100 also satisfies G’(x) = x2. Therefore, both F and G are antiderivatives of f.
Indeed, any function of the form H(x)=⅓ x3 + C, where C is a constant, is an antiderivative of f.
The question arises: Are there any others?
Here we see…
If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is
F(x) + Cwhere C is an arbitrary constant.
Theorem
- f - F
F is an antiderivative of f
Going back to the function f (x) = x2, we see that the general antiderivative of f is ⅓ x3 + C.
Antiderivatives of Power Functions
Fact (The Power Rule)•If f(x) = xr then f′(x) =rxr-1Example:f(x)=x2 then f’(x)?f’(x)= rxr-1
f’(x)= 2x2-1
f’(x)=2x
Rx2-1
By assigning specific values to C, we obtain a family of functions.
Their graphs are vertical translates of one another.
This makes sense, as each curve must have the same slope at any given value of x.
Family of Functions
Family of Functions
• The symbol ∫f(x)dx is traditionally used to represent the most general an antiderivative of f on an open interval and is called the indefinite integral of f .
Notation of Antiderivatives
• Thus, ∫f(x)dx means F’(x) = f (x).
Notation of Antiderivatives
The expression ∫f(x)dx reads:
“the indefinite integral of f with respect to x,” means to find the set of all antiderivatives of f.
Notation of Antiderivatives
( )f x dx
Integral sign
Integrand
x is called the variable of integration
Notation of Antiderivatives
Example: Notation of Antiderivatives
3 32 2because
3 3
x d xx dx C C x
dx
We can write:
Thus, we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).
Problem
Example -1
32
2
1Evaluate: x + dx
x
32
21
Solution: Let I = x + dxx
6 26 2
1 3= x + +3x + dx
x x
6 26 2
1 3x dx dx 3x dx dx
x x
6 -6 2 -2= x dx+ x dx+3 x dx+3 x dx 7 -5 3 -1x x 3x x
= + + +3× +C7 -5 3 -1
73
5x 1 3
= - +x - +C7 x5x
The final answer
Example -2
1Evaluate: dx
3x+1 - 3x - 11
Solution: Let I = dx3x+1 - 3x - 1
1 3x+1+ 3x - 1= × dx
3x+1 - 3x - 1 3x+1+ 3x - 1
3x+1+ 3x - 1
= dx3x+1 - 3x - 1
3x+1+ 3x - 1= dx
2
1 1= 3x+1 dx+ 3x - 1 dx
2 2
3 32 2
3x+1 3x - 11 1= + +C
3 32 2×3 ×32 2
3 32 2
1 1= 3x+1 + 3x - 1 +C
9 9
Example -2 Cont.
Example -3
xEvaluate: dx
x+2x
Solution: Let I = dxx+2
x+2- 2= dx
x+2x+2 1
= dx - 2 dxx+2 x+2
1 1
-2 2= x+2 dx - 2 x+2 dx
3 12 2
32
x+2 x+2= - 2 +C
12
322
= x+2 - 4 x+2 +C3
Example -4
2Let 5 7 then 10
duu x dx
x
Evaluate
3/ 21
10 3/ 2
uC
3/ 225 7
15
xC
25 7x x dx
2 1/ 215 7
10x x dx u du
Pick u, compute
du
Sub in
Sub in
Integrate
Example -5
Evaluate 4 1 x dxLet 4 1u x
4 du dx
1
4du dx
Solve for dx.
1
21
4
u du3
22 1
3 4u C
3
21
6u C
3
21
4 16
x C
Thank [email protected]
