(Indefinite) Integration “Add Math” Calculus (Indefinite) Integration

We first need to consider an example of differentiation e.g.1 Differentiate (a) (b) (a) Equal ! (b) The gradient functions are the same since the graph of is a just a translation of

Graphs of the functions e.g. the gradient at x = -1 is -2 At each value of x, the gradients of the 2 graphs are the same

C is called the arbitrary constant Indefinite integration is the reverse of differentiation If we are given the gradient function and want to find the equation of the curve, we reverse the process of differentiation BUT the constant is unknown So, C is called the arbitrary constant or constant of integration The equation forms a family of curves

e.g.2 Find the equation of the family of curves which have a gradient function given by Solution: To reverse the rule of differentiation: add 1 to the power divide by the new power

Tip: Check the answer by differentiating e.g.2 Find the equation of the family of curves which have a gradient function given by Solution: To reverse the rule of differentiation: add 1 to the power divide by the new power add C Tip: Check the answer by differentiating

The graphs look like this: The gradient function ( Sample of 6 values of C )

We can only find the value of C if we have some additional information e.g. 3 Find the equation of the family of curves with gradient function Solution: The index of x in the term 3x is 1, so adding 1 to the index gives 2. The constant -1 has no x. It integrates to -x. We can only find the value of C if we have some additional information

Exercises Find the equations of the family of curves with the following gradient functions: 1. 2. 3. N.B. Multiply out the brackets first

Exercises Find the equations of the family of curves with the following gradient functions: 1. 2. 3.

Finding the value of C e.g.1 Find the equation of the curve which passes through the point (1, 2) and has gradient function given by Solution:

6 is the common denominator Finding the value of C e.g.1 Find the equation of the curve which passes through the point (1, 2) and has gradient function given by Solution: (1, 2) is on the curve: 6 is the common denominator

Exercises 1. Find the equation of the curve with gradient function which passes through the point ( 2, -2 ) Find the equation of the curve with gradient function which passes through the point ( 2, 1 ) 2.

Solutions 1. Ans: ( 2, -2 ) lies on the curve

Solutions 2. ( 2, 1 ) on the curve So,

Notation for Integration e.g. 1 We know that Another way of writing integration is: Called the integral sign We read this as “d x ”. It must be included to indicate that the variable is x In full, we say we are integrating “ with respect to x “.

e.g. 2 Find (a) (b) Solution: ( Integrate with respect to x ) (a) (b) ( Integrate with respect to t ) e.g. 3 Integrate with respect to x Solution: The notation for integration must be written We have done the integration so there is no integral sign

Exercises 1. Find (a) (b) 2. Integrate the following with respect to x: (a) (b)

Summary Indefinite Integration is the reverse of differentiation. A constant of integration, C, is always included. Indefinite Integration is used to find a family of curves. To find the curve through a given point, the value of C is found by substituting for x and y. There are 2 notations:

Summary Indefinite Integration is the reverse of differentiation. A constant of integration, C, is always included. There are 2 notations: