**Theorem 5.5 Log Rule for Integration**

Let u be a differentiable function of x.

Because du = u’dx, the second formula can also be written
as

Ex 1 Find the indefinite integral.

Ex 2 Find the indefinite integral.

Ex 3 Find the indefinite integral.

Ex 4 Find the indefinite integral by u-substitution.
(Hint: Let u be the

denominator of the integrand .)

Ex 5 Find the indefinite integral. You may have to use
long division first .

**Guide lines for Integration **

1. Learn a basic list of integ ration formulas . (Including those given in

this section you now have 12 formulas: the Power Rule , the Log Rule,

and ten trigonometric rules . By the end of Section 5.7, this list will

have expanded to 20 basic rules.)

2. Find an integration formula that resembles all or part of the integrand,

and, by trial and error, find a choice of u that will make the integrand

conform to the formula.

3. If you cannot find a u- substitution that works , try altering the

integrand. You might try a trigonometric identity, multiplication and

division by the same quantity, or addition and subtraction of the same

quantity. Be creative.

4. If you have access to computer software that will find antiderivatives

symbolically , use it.

Ex 6 Find the indefinite integral:

**Integrals of the Six Basic Trigonometric Functions**

Ex 7 Solve the differential equation. Use a graphing
utility to graph three

solutions , one of which passes through the given point.
,(0,4)

Ex 8Evaluate the definite integral. Use a graphing utility
to verify your

result.

Ex 9 Find the area of the region bounded by the graphs of
the equations .

Use a graphing utility to verify your result.

Ex 10 Find the average value of the function over the
given interval.

5.2 Homework # 1 – 73 e.o.o. and 85, 87, 91