II. Integrating Improper Rational Functions

Although the method of partial fractions only applies to proper rational functions, an improper rational function can be integrated by performing long division (or synthetic division). If is a rational function where and are polynomial functions and the degree of is greater than or equal to the degree of , then by long division, where is a proper rational fraction. Since is a proper rational function, it can be decomposed into partial fractions.

Example 12: Find .

. Thus, .

.

4. Integration of irrational functions

Type 1.

Integrand contains  .

Use the substitution  .

Type 2.

Integral is of the form  .

Write   as .

Type 3.

Integrand contains  , or .

This was discussed in "trigonometric substitutions above". Here is a summary:

1. For  , use  .

2. For  , use  .

3. For  , use  .

Type 4.

Integral is of the form . 

Use the substitution  .

Type 5.

Other rational expressions with the irrational function  

1. If  , we can use .

2. If  , we can use  .

3. If   can be factored as  , we can use .

4. If   and   can be factored as  , we can use 

Frequently occuring integrals of irrational functions are: , where and is possitive on an interval. We can exclude the case, that the polynomial has a double root.

Taking the factor (if ) or (if ) we reduce the integral to the form or , leading (by means of substitution) to the integrals:

or , respectively.

5. Integration of trigonometric functions

1. Given an integral , i.e. the integrand is a rational function in terms of and . By the substitution the integral is reduced to an integral of a rational function. If , then , , and .

2. If = , then .

3. If = - , then .

If =- , then .

4. , т and п – even non-negative integers, then , .

5. For integrals we use following formulas:

5. , then and .

_{LECTURE 9-10.}

The definite integral. Properties of definite integrals. The Newton-Leibniz formula. Applications of definite integrals in economics. Calculation the arc length, the amount of body rotation. The improper integral. Applications of improper integrals in economics

1. Concept of definite integral

Definition 1. Let be a continuous function defined on divide the interval by the points from to into subintervals. (not necessarily equal width) such that when , the length of each subinterval will tend to zero.

In the i^th subinterval choose for . If exists and is independent of the particular choice of and , then we have .

Remark. For equal width, i.e. divide into equal subintervals of length, i.e. , we have .

Choose and

or .

2. Main properties of definite integrals

Properties of Definite Integrals

P1. The value of the definite integral of a given function is a real number, depending on its lower and upper limits only, and is independent of the choice of the variable of integration, i.e.

.

P2.

P3.

P4. Let , then .

P5. Comparison of two integrals

If , then

P6. Rules of Integration

If are continuous function on then

(a) for some constant k.

(b) .

Newton – Leibniz formula

Comparing the two formulas of the curvilinear trapezoid area, we make the conclusion: if  F ( x ) is primitive for the function  f ( x )  on a segment  [ a, b ] , then

This is the famous Newton – Leibniz  formula. It is valid for any function  f ( x ), which is continuous on a segment  [ a , b ] .

(i) If (Even Function)

then .

(ii) If (Odd Function)

Then .