BY DR. SAFA AHMED ELASKARY FACULTY OF ALLIED MEDICAL OF SCIENCES Lecture (1) Antiderivatives and the...
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Transcript of BY DR. SAFA AHMED ELASKARY FACULTY OF ALLIED MEDICAL OF SCIENCES Lecture (1) Antiderivatives and the...
BYDR. SAFA AHMED ELASKARY
FACULTY OF ALLIED MEDICAL OF SCIENCES
Lecture (1)
Antiderivatives and the Rules of Integration
A function F is an antiderivative of f on an interval I if F’(x)=f(x) for all x in I.
Let’s use an example to figure out what this statement means…
Suppose we know f(x) = 2x and we want to find its antiderivative, F.
If f(x) = 2x, then F’(x) = 2x. So we know the derivative of F.
Think backwards, what function has a derivative equal to 2x ?
F(x) = x² !!!
To find the antiderivative, do the reverse of finding the derivative.
Is F(x) = x² the only function whose derivative is 2x ? Or in otherwords, is F(x) = x² the only antiderivative of 2x ?
Theorem 1: Let G be an antiderivative of a function f. Then, every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant.
Example 1: Let F(x) = x² + 4 and let G(x) = x²- 1 Then F’(x) = 2x and G’(x)= 2x
Thus both F and G are antiderivatives of f(x) = 2x.Note two functions which have the same derivative will only differ by the constant.
This means the antiderivative of a function is a family of functions as pictured on the next slide.
So the answer to the question is F(x) = x²the only antiderivative of2x is NO!!
The Indefinite Integral
The process of finding all antiderivatives of a function is called antidifferentiation, or integration.
is the symbol used for integration and is called the integral symbol.
We write C)x(Fdx)x(f
This is called an indefinite integral, f(x) is called the integrand and C is called the constant of integration.
Basic Integration Rules
CkxkdxRule 1: (k, a constant)
Example 2:
Cx2dx2
Example 3:
Cxdx 22
Keep in mind that integration is the reverse of differentiation. What function has a derivative k?
kx + C, where C is any constant.
Another way to check the rule is to differentiate the result and see if it matches the integrand. Let’s practice.
Before we list Rule 2, let’s go back and think about derivatives.
When we used the power rule to take the derivative of a power, we multiplied by the power and subtracted one from the exponent.
Example: 23 x3xdxd
Since the opposite of multiplying is dividing and the opposite of subtracting is adding, to integrate we’d do the opposite. So, let’s try adding 1 to the exponent and dividing by the new exponent.
Check by differentiating the result:334 xx4
4x41
dxd
Since we get the integrand we know it works.
413
3 x41
13xdxx
Integrating:
Rule 2: The Power Rule C1nxx
1nn
n 1
Example 4: Find the indefinite integral dtt 3
Solution: C4tdtt
43
Example 5: Find the indefinite integral dxx23
CX52C
25
XC12
3xdxx 2
52512
3
23
Solution:
Basic Integration Rules
Example 6: Find the indefinite integral dxx13
Solution: C2x
1C2xC13
xxdxx1
2
2133
3
Example 7: Find the indefinite integral dx1Cxdx1 Solution:
Example 8: Find the indefinite integral dxx3 2
Solution: Cx3C1
3xC123xx3dx3x
11222
Here are more examples of Rule 1 and Rule 2.
Rule 3: The Indefinite Integral of a Constant Multiple of a Function
constantaiscf(x)dx,ccf(x)dx
Rule 4: The Sum Rule (or difference)
g(x)dxf(x)dxdxxgxf
g(x)dxf(x)dxdxxgxf
Rule 5: Cedxe xx
Rule 6: Cxlndxx1
Basic Integration Rules
To check these 2 rules, differentiatethe result and you’ll see that it matchesthe integrand.
Example 9: Integrate. dxe3xx3
x1x2 x
2 )(
Using the sum rule we separate this into 5 problems.
dxe3dxxdxx3dxx
1xdx2 x2
Call them: 1 2 3 4 5For 1 we will use rule 3 to bring the constant outside of the integral sign.
xdx2 Next we will use rule 2, the power rule to integrate.
2211
x2x211
x2xdx2
1
dxe3dxxdxx3dxx
1xdx2 x2
Call them: 1 2 3 4 5
2xxdx2
For 2 we will use Rule 6 the natural log rule.2
xdxx1 ln
For 3 we will first rewrite then use the constant rule (Rule 3) and then the power rule (Rule 2).
3
x3
1x312
x3dxx3dxx3 112
22
Example 9 continues…
1
dxe3dxxdxx3dxx
1xdx2 x2
Call them: 1 2 3 4 5
2x2xdx
For 4 we will rewrite and then use the power rule (Rule 2).
xlndxx1
For 5 we will use the constant rule (Rule 3) and then Rule 5 for ex.
x3dx
x32
4
2312
1
21
x32
121xdxxdxx
5xxx e3dxe3dxe3
Example 9 continues…
1 2 3
dxe3dxxdxx3dxx
1xdx2 x2
Call them: 1 2 3 4 5
2xxdx2
x
dxx1
ln
x3dxx
32
23
x32
dxx
x
x
e3dxe3
Ce3x32
x3xxdxe3dxxdxx
3dxx1xdx2 x2
32x
2 ln
So in conclusion:
You may be wondering why we didn’t use the C before now. Let’s say that we had five constants . Now we add all of them together and call them C. In essence that’s what’s going on above.
54321 CCCCC
Example 9 continues…
1 2 3 4 5
Here are some for you to try:
1. Integrate and check your answer by taking the derivative.
dxex3 x4
Click the correct answer below.
Ce5x3x x5
Ce4x3x x3
Cex513x x5
Sorry that’s not correct.
Think about the power rule for integration. You should add one to the exponent and divide by the new exponent.
Try again. Return to the previous slide.
Good Work!!
Here is the solution in detail.
Cex513x
Ce14x3x
dxedxx3dxdxex3
x5
x14
x4x4
2. Integrate.
dxx3
x142x 2
Click on the correct first step.
Cx3lnx14xx.separatelyfactoreachIntegrate 2
dxx4
x106terms.likethecombineand
togetherradicandtheinfactorstwotheMultiply
2
Be careful !!
Rule 4 states:
g(x)dxf(x)dxdxxgxf
g(x)dxf(x)dxdxxgxf
This does NOT apply to multiplication or division.
You should multiply the factors in the integrand, simplify and then use Rule 4 to integrate the terms.
dxx4
x106dxx
12x46x
2terms.likethecombineand
togetherradicandtheinfactorstwotheMultiply
22
Cx4x10ln6x
C1x4x10ln6x
dxx4dxx1106dx
.separatelytermeachintegrateto4RuleuseNext
1
2
Let’s look at the solution to the problem:f’(x)= 3x2 - 4x + 8f(1)= 9
Solution: First integrate both sides:
Cx82x4
3x3xf
23)(
Simplify: Cx8x2xxf 23 )(
Now find C by using the initial condition.Substitute 1 for x and 9 for f(x)
C2C79
C8219C181219 23
This gives the particular solution.
2x8x2xxf 23 )(
Review - Basic Integration Rules
Rule 1: (k, a constant) Ckxkdx
Rule 2: The Power Rule C1nxx
1nn
Rule 3: The Indefinite Integral of a Constant Multiple of a Function
constantaiscf(x)dx,ccf(x)dx
Rule 4: The Sum Rule (or difference)
g(x)dxf(x)dxdxxgxf
g(x)dxf(x)dxdxxgxf
Rule 5: Cedxe xx
Rule 6: Cxlndxx1