RULES FOR INTEGRATION
Although
integration is the inverse of differentiation and we were given rules for
differentiation, we are required to determine the answers in integration by
trial and error. However, there are some rules to aid us in the determination
of the answer.
In
this section we will discuss four of these rules and how they are used to
integrate standard elementary forms. In the rules we will let u and v denote a
differentiable function of a variable such as x. We will let C, n, and a denote constants.
Our
proofs will involve searching for a function F(x) whose derivative is
.
The integral of a differential of a function is the
function plus a constant.
PROOF: If
then
and
EXAMPLE. Evaluate the integral
SOLUTION: By Rule 1, we have
A constant may be moved across the integral sign. NOTE: A variable may NOT be
moved across the integral sign.
PROOF: If
then
and
EXAMPLE: Evaluate the integral
SOLUTION: By Rule 2,
and by Rule 1,
therefore,
The integral of
du may be obtained by adding 1 to the exponent
and then dividing by this new exponent. NOTE: If n is minus 1,
this rule is not valid and another method must be used.
PROOF. If
_{
}
then
EXAMPLE: Evaluate the integral
SOLUTION: By Rule 3,
EXAMPLE: Evaluate the integral
SOLUTION: First write the integral
as
Then, by Rule 2,
and by Rule 3,
The integral of a sum is equal to the sum of the
integrals.
PROOF: If
then
such that
where
EXAMPLE: Evaluate the integral
SOLUTION: We will not combine 2x and 5x.
where C is the sum of
.
EXAMPLE: Evaluate the integral
SOLUTION:
Now we will discuss the evaluation of the constant of
integration.
If we are to find the equation of a curve whose first derivative
is 2 times the independent variable x, we may write
or
We
may obtain the desired equation for the curve by integrating the expression for
dy; that is, by integrating both
sides of equation (1). If
then,
But,
since
and
then
We
have obtained only a general equation of the curve because a different curve
results for each value we assign to C. This is shown in figure 67. If
we specify that
x=0
And
y=6
we
may obtain a specific value for C and hence a particular curve.
Suppose
that
then,
or
C=6
Figure 67.Family of curves.
By
substituting the value 6 into the
general equation, we find that the equation for the particular curve is
which
is curve C of figure 67.
The
values for x and y will determine the value for C and also determine the
particular curve of the family of curves.
In
figure 67, curve A has a constant equal to  4, curve B has a constant equal
to 0, and curve C has a constant equal to 6.
EXAMPLE: Find the equation of the curve if its first derivative is 6 times the independent
variable, y equals 2, and x equals 0.
SOLUTION. We may write
or
such
that,
Solving
for C when
x=0
and
y=2
We
have
or
C=2
so
that the equation of the curve is
