24: Indefinite Integration © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

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

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

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

Indefinite Integration Make the power 1 more To reverse the rule of differentiation: e.g.2 Find the equation of the family of curves which have a gradient function given by Drop it thro` the trap door Solution:

Indefinite Integration Make the power 1 more To reverse the rule of differentiation: e.g.2 Find the equation of the family of curves which have a gradient function given by Drop it thro` the trap door add C Tip: Check the answer by differentiating Solution: Make the power 1 more and drop it through the trap door

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

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

Indefinite Integration 1. Find the equations of the family of curves with the following gradient functions: Exercises 2.3. N.B. Multiply out the brackets first

Indefinite Integration 1. Find the equations of the family of curves with the following gradient functions: Exercises 2.3.

Indefinite Integration Make the power 1 more Drop it thro` the trap door add C Reminder: n does not need to be an integer BUT notice that the rule is for It cannot be used directly for terms such as

Indefinite Integration e.g.1 Evaluate Solution: Using the law of indices, So, This minus sign makes the term negative.

Indefinite Integration e.g.1 Evaluate Solution: Using the law of indices, So, But this one... is an index

Indefinite Integration e.g.2 Evaluate We need to simplify this “piled up” fraction. Multiplying the numerator and denominator by 2 gives We can get this answer directly by noticing that dividing by a fraction is the same as multiplying by its reciprocal. ( We “flip” the fraction over ). Solution:

Indefinite Integration e.g.2 Evaluate We need to simplify this “piled up” fraction. Multiplying the numerator and denominator by 2 gives We can get this answer directly by noticing that... Solution:... dividing by a fraction is the same as multiplying by its reciprocal. ( We “flip” the fraction over ).

Indefinite Integration e.g.3 Evaluate Solution: So, Using the law of indices,

Indefinite Integration e.g.4 Evaluate Solution: Write in index form Split up the fraction Use the 2 nd law of indices: We cannot integrate with x in the denominator.

Indefinite Integration e.g.4 Evaluate Solution:Instead of dividing by,multiply by and The terms are now in the form where we can use our rule of integration.

Indefinite Integration Evaluate Exercise Solution: 1. 2.