Finding Antiderivatives

The inverse relationship between the operations of integration and differentiaion can be shown symbolically as follows.

d d x [ ∫ f ( x ) d x ] = f ( x )		Differentiation is inverse of integration
Integration is inverse of differentiation		∫ f ′ ( x ) d x = f ( x ) + C

This inverse relationship between integration and differentiation allow us to obtain integration formulas directly from differentiation formulas. In the following summary we list the integration formulas that correspond to some of the differentiation formulas we have studied up to this point.

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

REMARK
Be sure you see that the Simple Power Rule has the restriction that n cannost be -1. This means that we cannot use the Simple Power Rule to evaluate the integral ∫ 1 x d x .
We will be able to evaluate this integral after natural logarithmic function is introduce in the Second Level - The General Power Rule.

Applications of these rules are demonstarted in the following examples.

EXAMPLE 1
Find the indefinite integral ∫ 3 x d x .

SOLUTION

∫ 3 x d x = 3 ∫ x d x Constant Multiple Rule

= 3 ∫ x 1 d x Rewrite x = x 1

= 3 ( x 2 2 ) + C Power Rule ( n=1)

= 3 2 x 2 + C Simplify

When finding indefinite integrals, a strict application of the basic integration rules tends to produce cumbersome constants of integration. For instance, in Example 1, we could have written

∫ 3 x d x = 3 ∫ x d x = 3 ( x 2 2 ) + C = 3 2 x 2 + 3 C

However, since C represent any constant, it is both cumbersome and unnecessary to write 3C as the constant of ontegration, and we opt for the simpler 3 2 x 2 + C .
In Example 1, note that the general pattern of integration is similar to that of differentiation.

Given integral

&to;

Rewrite

&to;

Integrate

&to;

Simplify

This pattern is followed in the next example.

EXAMPLE 2
Apply the basic integration rules

Given integral	Rewrite	Integrate	Simplify

∫ 1 x 3 d x	∫ x - 3 d x	x - 2 - 2 + C	- 1 2 x 2 + C
∫ x d x	∫ x 1 / 2 d x	x 3 / 2 3 / 2 + C	2 3 x 3 / 2 + C

You can look for the more examples and test your ability for solving these problems.

REMARK
Remember that you can check your answer to an antidifferentiation problem by differentiating. For instance, in the previous Example, you can check that 2 3 x 3 / 2 is correct antiderivative by differentiating to obtain

d d x [ 2 3 x 3 / 2 ] = ( 2 3 ) ( 3 2 ) x 1 / 2 = x

∫ 3 x d x	= 3 ∫ x d x	Constant Multiple Rule
	= 3 ∫ x 1 d x	Rewrite x = x 1
	= 3 ( x 2 2 ) + C	Power Rule ( n=1)
	= 3 2 x 2 + C	Simplify