MAE101 CAL v2 chapter 6 techniquse of integration

Chapter 3TECHNIQUES OF INTEGRATION Page 261-330, Calculus Volume 2 Dr.. 3.1 Integration by Parts 3.6 Numerical Intergration 3.7 Improper IntegralsCONTENT... 3.1 Integration by PartsIn th

Chapter 3

TECHNIQUES OF INTEGRATION

(Page 261-330, Calculus Volume 2)

Dr Tran Quoc Duy

Email: duytq4@fpt.edu.vn

3.1 Integration by Parts 3.6 Numerical Intergration 3.7 Improper Integrals

CONTENT

3.1 Integration by Parts

In this section, we will learn:

How to integrate complex functions by parts.

TECHNIQUES OF INTEGRATION

Thus, by the Substitution Rule, the formula for integration

Evaluating both sides of Formula 1 between a and b, assuming

f’ and g’ are continuous, and using the FTC, we obtain:

Evaluate I= ∫ e x sinx dx

 e x does not become simpler when differentiated

 Neither does sin x become simpler.

Example 2

INTEGRATION BY PARTS

Nevertheless, we try choosing

u = e x and dv = sin x

 Then, du = e x dx and v = – cos x.

Example 2

INTEGRATION BY PARTS

So, integration by parts gives:

The integral we have obtained, ∫ e x cos x dx,

is no simpler than the original one

 At least, it’s no more difficult

 Having had success in the preceding example integrating by parts twice, we do it again

Example 2

INTEGRATION BY PARTS

This time, we use

u = e x and dv = cos x dx

Then, du = e x dx, v = sin x, and

we get:

 This can be regarded as an equation to be

solved for the unknown integral

Adding to both sides ∫ e x sin x dx,

we obtain:

Example 2

2  ex sin x dx = − ex cos x e + x sin x

INTEGRATION BY PARTS

Dividing by 2 and adding the constant

of integration, we get:

Example 2

1 2

ò

INTEGRATION BY PARTS

Numerical Integration

Left endpoint Method

Right endpoint Method

f x

f x dx

x

Trapezoidal Method

)] (

) (

[ 2

)]

( )

(

[ 2

f

x x

f x

f

x dx

x



)](

)(

2

)(2)

([ f x0 f x1 f x n 1 f x n



Simpson Method

)] ( )

( 4 ) (

[

)]

( )

( 4 ) ( [ )

(

b

x f x

f x

f

x x

f x

f x

f

x dx

x



Estimate error for Midpoint and Trapezoidal method

• Suppose | f’’(x) | ≤ K for a ≤ x ≤ b.

• If E T and E M are the errors in the Trapezoidal and

Midpoint Rules, then

Estimate error for Simpson method

• Suppose | f(4)(x) | ≤ K for a ≤ x ≤ b.

• If E S is the error in the Simpson method, then

5 4

180

s

K b a E

n

-£

Example

How large should we take n in order to guarantee

that the Trapezoidal, Midpoint Rule, Simpson rule

3 2

2(1)

0.0001

12n <

| f ’’ (x) | ≤ 2 for 1 ≤ x ≤ 2

Accuracy to within 0.0001 means that error < 0.0001

Trapezoidal: Choose smallest n so that:

24(1)

0.0001

180n <

3.7 Improper Integrals

In this section, we will learn:

How to solve definite integrals where the interval is infinite and where the function has an infinite discontinuity

TECHNIQUES OF INTEGRATION

If exists for every number t ≥ a, then

provided this limit exists (as a finite number)

If exists for every number t ≤ a, then

provided this limit exists (as a finite number)

f x dx f x dx

− = →−

CONVERGENT AND DIVERGENT

The improper integrals and

are called:

 Convergent if the corresponding limit exists

 Divergent if the limit does not exist

If both and are convergent,

For what values of p is the integral convergent?

¥

ò

Example 1



dx e

x2 x3

If f is continuous on [a, b) and is discontinuous at b,

If f is continuous on (a, b] and is discontinuous at a, then

if this limit exists (as a finite number)

The improper integral is called:

 Convergent if the corresponding limit exists

 Divergent if the limit does not exist.

If f has a discontinuity at c , where a < c < b, and

both and are convergent, then we

) (

Answer: diverges if p 1 and converges if p<1.

IMPROPER INTEGRAL OF TYPE 2

Suppose f and g are continuous functions with f(x) ≥ g(x) ≥ 0 for x ≥

| ) cos(

| 0

x x

Does converge? = 

1

2

| ) cos(

|

x

dx x

I