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

Indefinite Integration Module C1 AQA Edexcel OCR MEI/OCR Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

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

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

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

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

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

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

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

Indefinite Integration 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 “.

Indefinite Integration e.g. 2 Find (a) (b) 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 (b) ( Integrate with respect to t ) ( Integrate with respect to x ) (a)

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

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

Indefinite Integration

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

Indefinite Integration We can only find the value of C if we have some additional information ( Sample of 6 values of C ) The gradient function

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

Indefinite Integration Solution: The index of x in the term 3x is 1, so adding 1 to the index gives 2. e.g. 2 Find the equation of the family of curves with gradient function The constant  1 has no x. It integrates to  x.

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

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