**Integration** :Is the reverse process of
differentiation, i.e. the process of finding the expression for y in
terms of x when given the gradient function.

The symbol for integration is

This is the general power of integration it works for all values of n except for n = -1

**Example**

1.

2. Integrate the following with respect to x

(i)3x^{2}

Solution

** Integration of constant**

**Integration by change of variables**

If x is replaced by a linear function of x, say of the form ax + b, integration by change of variables will be applied

E.g

**Example**

**Note:** 2x is the derivative of x^{2} + 1 in this case substitution is useful

i.e. let u = x^{2} + 1

This converts into the form

**Standard integrals**

·

**EXERCISE**

Find the integral of the following functions

**gration by partial fraction**

Integration by partial fraction is applied only for proper fraction

E.g.

Note that:

**Improper fraction**

If the degree of numerator is equal or greater than of denominator, adjustment must be made

**Example**

1. Find

Solution

If the denominator doesn’t factorize, splitting the numerator will work

→ Numerator = A (derivative of denominator) + B

**Example**

Solution

**Important**

It can be shown that

**EXERCISE**

**Integrated of the form**

**Note that:**

1. If the denominator has two real roots use partial fraction

2. If the denominator has one repeated root use change of variable or recognition

3. If the denominator has no real roots, use completing the square

**Integration of Trigonometric Expression
**Integration of Even power of

**Note that**

For integrand containing and , or even powers of these, the change of variable can be used.

**APPLICATION OF INTEGRATION**

To determine the area under the curve

= f (b) – f (a)

**Examples**

1. Find the area under the curve f(x) =x^{2}+1 from x=0 to x=2

2. Find the area under the curve f(x) = from x=1 to x=2

3. Find the area bounded by the function f(x) =x ^{2}-3, x=0, x=5 and the x- axis

**Solution**

- f(x) = + 1

y intercept=1

**EXERCISE**

1. Find the area between y = 7-x^{2 }and the x- axis from x= -1 to x=2

2. Find the area between the graph of y=x^{2 }x – 2 and the x- axis from x= -2 to x=3

**Solution**

^{1. }y =7-x^{2}

Where y- intercept =7

= 6.67 + 11.3

=17.97sq units

**Volume of the Solids of Revolution**

The volume,V of the solid of revolution is obtained by revolving the
shaded portion under the curve, y= f(x) from x= a to x =b about the x
-axis is given by

**Example 1**

Find the volume of revolution by the curve y=x^{2} from x=0 to x=2 given that the rotation is done about the the x- axis

**Exercise**

1. Find the volume obtained when each of the regions is rotated about the x – axis.

a) Under y= x^{3}, from x =0 to x=1

b) Under y^{2}= 4-x, from x=0 to x=2

c)Under y= x^{2}, from x=1 to x=2

d)Under y= √x, from x=1 to x=4

2. Find the volume obtained when each of the region is rotated about the y-axis.

a) Under y= x^{2}, and the y-axis from x=0 to x=2

b) Under y= x^{3}, and the y-axis from y=1 to y=8

c) Under y= √x, and the y-axis from y=1 to y=2

** LENGTH OF A CURVE**

Consider the curve

Example

Find the length of the part of the curves given between the limits:

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