Inverse substitution rule

1. Inverse substitution rule • Inverse Substitution Rule If and is differentiable and invertible. Then

2. Example: trigonometric substitutions • Ex. Evaluate • Sol. Let then From we derive

3. Example: trigonometric substitutions • Ex. Evaluate • Sol. Let then From we derive

4. Example • Ex. Evaluate • Sol.

5. Example • Ex. Evaluate substitution • Ex. Find substitution • Ex. Find • substitution

6. Example: reciprocal substitution • Ex. Evaluate • Sol. Let then

7. Example: rational substitution • Ex. Evaluate • Sol. Let then

8. Inverse substitution for definite integral • The Inverse Substitution Rule for definite integral: If x=g(t) is differentiable, invertible and, when x is in between a and b, t is in between and Then

9. Example • Ex. Evaluate • Sol. Let then and when x changes from a to 2a, t changes from 0 to

10. Example • Ex. Evaluate • Sol. Since let Then and when x changes from 2 to 3, t changes from 0 to

11. Example: application of substitution • Ex. Find • Sol. Let then

12. Example: application of substitution • Ex. Find the definite integral • Sol. By substitution

13. Integration of rational functions • Any rational function can be integrated by the following two steps: a). express it as a sum of simpler fractions by partial fraction technique; b. integrate each partial fraction using the integration techniques we have learned. • For example, since we have

14. Technique for partial fraction • Take any rational function where P and Q are polynomials. • If the degree of P is less than the degree of Q, we call f a proper fraction. • If f is improper, that is, degree of P greater than or equal to degree of Q, then (by long division) we must have where S and R are also polynomials and degree of R less than degree of Q.

15. Technique for partial fraction • For example, by long division, we can derive • For the above reason, we only need to consider the proper rational functions. • The next step is to factor the denominator Q(x). It can be shown that any polynomial can be factored as a product of linear factors (in the form ax+b) and irreducible quadratic factors (in the form ). For example,

16. Technique for partial fraction • The third step is to express the proper rational function R(x)/Q(x) as a sum of partial fractions of the form • These two kind of rational functions can be integrated as

17. Example • Ex. Find • Sol.

18. Technique for partial fraction • From the above analysis, we see that how to split a rational function into partial fractions is the key step to integrate the rational function. • When Q(x) contains factor the partial fractions contain • When Q(x) contains irreducible factor the partial fractions contain

19. Example • Ex. Find • Sol. Since the partial fraction has the form Expanding the right side and comparing with the left side,

20. Example • Ex. Find • Sol.

21. Remark • There are two methods to find the coefficients in the partial fractions. One method is comparing the corresponding coefficients of polynomials on both sides; the other is taking some special values of x in the identity. • For instance, in the last example, we have

22. Example • Ex. Find • Sol.

23. Homework 18 • Section 7.2: 18, 24, 42, 44 • Section 7.3: 5, 6, 24, 27