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

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

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

Example Ex. Evaluate Sol.

Example Ex. Evaluate substitution Ex. Find substitution Ex. Find substitution

Example: reciprocal substitution Ex. Evaluate Sol. Let then

Example: rational substitution Ex. Evaluate Sol. Let then

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

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

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

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

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

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

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.

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,

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

Example Ex. Find Sol.

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

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

Example Ex. Find Sol.

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

Example Ex. Find Sol.

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