We say again the integration formulas that corespond to some of the differentiaion formulas.

Basic Integration Rules
1. Constant Rule ∫ k   d x = k x + C ,
k is a constant
2. Constant Multiple Rule ∫ k f ( x )   d x = k ∫ f ( x )   d x
3. Sum Rule ∫ [ f ( x ) ± g ( x ) ]   d x = ∫ f ( x )   d x ± ∫ g ( x )   d x
4. Simple Power Rule ∫ x n   d x = x n + 1 n + 1 + C , n &neq; - 1

EXAMPLE 1
Find the following indefinite integral.
∫ 1 d x

SOLUTION
By the Constant Rule we have
∫ 1 d x = x + C
REMARK
This integral is usually simplified to the form
∫ 1 d x = ∫ d x


EXAMPLE 2
Find the following indefinite integral.

∫ ( x + 2 ) d x

SOLUTION
Using the Sum Rule, we integrate each part separately.
∫ ( x + 2 ) d x = ∫ x d x + ∫ 2 d x = x 2 2 + 2 x + C


EXAMPLE 3
Find the following indefinite integral.

∫ ( 3 x 4 - 5 x 2 + x ) d x

SOLUTION
Try to identify each basic integration rule used to evaluate this integral.
∫ ( 3 x 4 - 5 x 2 + x ) d x = ∫ ( 3 x 4 ) d x - ∫ ( 5 x 2 ) d x + ∫ ( x ) d x
       = 3 ( x 5 5 ) - 5 ( x 3 3 ) + ( x 2 2 ) + C


EXAMPLE 4 - Rewriting Before Integrating
Find the indefinite integral.

∫ x + 1 x d x

SOLUTION
We begin by rewriting the quotient in the integrand as a sum.
∫ x + 1 x d x = ∫ ( x x + 1 x ) d x

Then, rewriting each term using fractional exponents, we obtain
∫ ( x x + 1 x ) d x = ∫ ( x 1 / 2 + x - 1 / 2 ) d x
       = x 3 / 2 3 / 2 + x 1 / 2 1 / 2 + C
       = 2 3 x 3 / 2 + 2 x 1 / 2 + C

REMARK
When integrating quotients, don§t make the mistake of integrating the numerator and the denominator separately. This is no more valid in integration than it is in differentiation. For instance, in Example 4 be sure you se that
∫ x + 1 x d x &neq; ∫ ( x + 1 ) d x ∫ x d x
You can look for the more examples and test your ability for solving these problems.