**INTEGRATION**

**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 , denote the integrate of a function with respect to x

If

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**

The result for differentiating c x is c

Properties

(1)

(2)

**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.

Considering in similar way gives the general result

**Example**

Find the integral of the following

a) (3x – 8) ^{6} b)

Solution (a)

Solution (b)

→ If

** **

**Example**

1. Find

Solution

2. Find

Solution

Integration of exponential function

**Example 01**

Solution

**Alternative**

**Example 02**

Solution

Alternative

**Integrating fraction**

If

Differentiating with respect to x gives

**Example**

1. ,given that f(x)=x^{2}+1

Solution

2. Find

solution

**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**

·

·

·

·

· →∫sec x tan xdx=sec x+c

·

·

·

·

·

·

·

·

**EXERCISE**

Find the integral of the following functions

i)

ii)

iii)

iv)

**Integration by partial fraction**

Integration by partial fraction is applied only for proper fraction

E.g.

Note that:

The expression is not in standard integrals

**Example 01**

**Example 02**

**Improper fraction**

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

**Example**

1. Find

Solution

Both numerator and denominator have the degree of 2

2.

3.

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**

I.

II.

III.

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

E.g.

I.

II.

III.

**Integral of the form**

**Example
** →

Then hyperbolic function identities is identities is used

**Note that:**

If the quadratic has 1 represented root, it is easier

E.g.

**EXERCISE**

Find the following

i.

ii.

iii.

iv.

v.

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

**Note that:**for even power of use the identity

i)

ii)

**Example 01**

Find

**Example 02**

** Odd powers of **

For odd powers of use identity

**Example**

Find

** Any power of tan**

The identity is useful as it is the fact that It will be understood that;

**Example:**

1. Find

** Solution:**

2.

**solution**

**Multiple Angles**

To integrate such type of integral, one of the factor formulae will be used

1.

2.

3.

4.

**Example**

1. Find

**Solution**

2.

Solution

**EXERCISE**

Find the integral of the following

1.

2.

3.

4.

Integrated by change of variables

1.

**Note that**

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

Example

**APPLICATION OF INTEGRATION**

To determine the area under the curve

Given A is the area bounded by the curve y=f(x) the x -axis and the line x=0 and x=b where b> a

The area under that curve is given by the define definite integral of f(x) from a to b

= 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: