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