Chapter 1

Indefinite Integration

1.1 Indefinite Integrals

Indefinite integration may be regarded as the inverse operation to differentiation. This means that the derivative of an indefinite integral of a function is the function itself.

Definition: Suppose . Then is said to be an indefinite integral or anti-derivative of . This is written as

Note that, if C is any constant, the derivative of the right hand side is , since the derivative of a constant is 0. Such a constant is called a constant of integration. The function is called the integrand.

Example:

since

since Exercise:

Verify the following formulas by differentiating

(a) ,

(b) ,

For ease of reference, Table below gives the indefinite integrals of some of the commonly occurring functions whose validity can be established by taking the derivative of the right hand side. In this Table a is a constant and C is the arbitrary constant of integration. Table of Elementary Integrals

1. 2. 3.

4. 5.

6. 7.

Note that . The domain of is x > 0 and is . The integration include positive and negative values of x, . It can be verified that

if x < 0, then letting , y > 0, we have

Thus for both cases, x > 0 or x < 0, we have and hence

Theorem: Suppose and are functions whose integrals exist. Then

This follows immediately by differentiation.

Example:

Exercise 1.1 :1. Find the following integrals:

(a) (b) (c)

(d) (e) (f)

(g) (h) (i) (j)

2. Find the function f such that and .

3. Find the function y such that and .

4. Find the function f such that and such that and

Answers:

1. (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

2.

3.

4.

1.2 Method of Integration

1.2.1 Reduction to Elementary Form

Some functions can be reduced, by elementary manipulations, to forms that we know how to integrate. Important Elementary Trigonometric formulae

1. 2.

Example:

Exercise 1.2.1:1.Evaluate the following integrals:

(a) (b) (c) (d)

(e) (f) (g) (h)

Answers:

1. (a)

(b)

(c)

(d)

(e) (f)

(g)

(h)

1.2.2 Integration by Substitution

This technique depends on the following result:Suppose G and u have continuous derivatives. Then

Note that we can formally replace by

Example: (a) Evaluate

let,

Then Which is summarized as

The above formula is used to evaluate the other integrals frequently.

(b)

Exercise 1.2.2: 1.Evaluate the following integrals by making the indicated substitutions:

(a) (b)

(c) (d)

(e) (f) 2. Evaluate the following integrals.

(a) (b) (c)

(d) (e) (f)

Answers: 1.

(a)

(b)

(c)

(d)

(e)

(f) 2.

(a)

(b)

(c)

(d)

(e)

(f)

1.2.3 Integration by Parts

Suppose that u and v are differentiable. From the product rule

.Integrating with respect to x and transposing, we have

.This can also be expressed as

.Either of the above formulae is known as the integration by parts formula. This formula is useful in integrating the product of two functions.Note:Care must be taken in the selection of u and dv/dx. The wrong choice may lead to a more complicated integral than the original one.

Examples(a) Evaluate .

Setting (because its derivative is simpler and of lower degree than x)

and which gives , we have

.

(b) Evaluate .

Integrating by parts using or and or we obtain

=

(integrating by parts again)

Exercise1.2.3: 1.Evaluate the following integrals:

(a) (b) (c) (d)

(e) (f) (g) (h) Answers: 1. (a)

(b)

(c)

(d) (e)

(f)

(g)

(h)

3