“Teach A Level Maths” Vol. 1: AS Core Modules 50: Harder Indefinite Integration © Christine Crisp

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n does not need to be an integer BUT notice that the rule is for Reminder: add 1 to the power divide by the new power add C n does not need to be an integer BUT notice that the rule is for It cannot be used directly for terms such as

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

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

e.g.2 Evaluate Solution: 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 ).

e.g.2 Evaluate Solution: 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 ).

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

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

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

( 1, 0 ) on the curve: e.g.5 The curve passes through the point ( 1, 0 ) and Find the equation of the curve. Solution: It’s important to prepare all the terms before integrating any of them ( 1, 0 ) on the curve: So the curve is

Exercise Evaluate 1. 2. Solution:

( 2, 0 ) on the curve: Exercise 3. Given that , find the equation of the curve through the point ( 2, 0 ). Solution: ( 2, 0 ) on the curve: So the curve is

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e.g.1 Evaluate Solution: Using the law of indices, So, This minus sign . . . . . . makes the term negative. But this one is an index

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:

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

e.g.4 Evaluate Solution: Write in index form Split up the fraction We cannot integrate with x in the denominator. Use the laws of indices: and

The terms are now in the form where we can use our rule of integration.

( 1, 0 ) on the curve: e.g.5 The curve passes through the point Solution: e.g.5 The curve passes through the point ( 1, 0 ) and . Find the equation of the curve. ( 1, 0 ) on the curve: So the curve is It’s important to prepare all the terms before integrating any of them