Rules
Integrals of the Squares of sin x  and  cos x
Integration by Substitution
Power Rule



Rules for Indefinite Integration
1.] Constant Multiple Rule

where is a constant with respect to .
We can move a constant that is factor to a position outside of the integral sign.

Examples
a)


b) But in

the can not be moved outside of the integral sign because the is not
a factor in an appropriate product, an appropriate product being a product of
the form  
.

c) In

you can do algebra to rewrite the integrand so that it is in the appropriate product form.
.
The is a constant and a factor in a procduct.
We can use the constant multiple rule for integrals.



2.] Rule for Negatives

This is just Rule 1 with .


3.] Sum and Difference Rule

We can distribute the integral sign across sums and differences.


Example
d)








 is a constant.
We can simplify the expression by letting

and writing the final answer as

.

From now on, when possible, let's do this when we integrate.

See Examples 1 - 2, pages 323 - 323.


The Integrals of  sin²x  and  cos²x





See Example 3, pages 324 - 325.

Integration by Substitution
The integral formula (Table 4.1, page 315)

applies only when the integrand is precisely .
It does not directly apply to


because what is being raised to the power 7 is not , but .
Also, there is the factor , a factor which does not appear in the
simple integral formula.
The integrand in the formula is just .
However, the simple formula can be generalized to include this more
complicated integrand..
By the Chain Rule, if is a function of ,
,
then


                      .
Rewriting this in the context of antiderivatives, it becomes
.
If we think of the differentials on the left side of the equation cancelling, we get
the General Power Rule in integral differential form
         .

This is the form of the Power Rule we need to work out
.
If we let , then , so the integral is just
.
Back in terms of , this is
.

The formulas in Table 4.1 involving similarly can be generalized by
replacing by and by .
For example,

generalizes to



Example
Find
.
Let , then .
The integrand already contains , we only need a factor in front of it
to have the combination
.
is a constant.
We can move constants back and forth across the integral sign
by the Constant Multiple Rule.
Multiply by and divide by to get the form we want.
Nothing, except the form of the integrand has changed, because
.



          

            .

This is the final answer in terms of .