1. PROGRAMME F13
INTEGRATION

2. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

3. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

4. Integration
Constant of integration
Integration is the reverse process of differentiation. For example:
The integral of 4x3 is then written as:
Its value is, however:

5. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

6. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

7. Standard integrals
Just as with derivatives we can construct a table of standard integrals:

8. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

9. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

10. Integration of polynomial expressions
Just as polynomials are differentiated term by term so they are integrated, also term by term. For example:

11. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

12. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

13. Functions of a linear function of x
To integrate
we change the variable by letting u = ax + b so that du = a.dx. Substituting into the integral yields:

14. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

15. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

16. Integration by partial fractions
To integrate we note that
so that:

17. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

18. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

19. Areas under curves
Area A, bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b, is given by:
where

20. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

21. Integration
Standard integrals
Integration of polynomial expressions
Functions of a linear function of x
Integration by partial fractions
Areas under curves
Integration as a summation

22. Integration as a summation
Dividing the area beneath a curve into rectangular strips of width x gives an approximation to the area beneath the curve which coincides with the area beneath the curve in the limit as the width of the strips goes to zero.

23. Integration as a summation
If the area is beneath the x-axis then the integral is negative.

24. Integration as a summation
The area between a curve an intersecting line
The area enclosed between y1 = 25 – x2 and y2 = x + 13 is given as:

25. Learning outcomes
Appreciate that integration is the reverse process of differentiation
Recognize the need for a constant of integration
Evaluate indefinite integrals of standard forms
Evaluate indefinite integrals of polynomials
Evaluate indefinite integrals of ‘functions of a linear function of x’
Integrate by partial fractions
Appreciate the definite integral is a measure of an area under a curve
Evaluate definite integrals of standard forms
Use the definite integral to find areas between a curve and the horizontal axis
Use the definite integral to find areas between a curve and a given straight line