Integration L.O. All pupils understand what integration is All pupils can integrate individual terms

Recap of differentiation If I wanted to differentiate the following what should I do? 𝑦= 𝑥 2 3) 𝑦= 2 𝑥 5 +3 𝑥 7 𝑥 3 1) 𝑦= 1 2 𝑥 2 +5𝑥 2) 𝑦= 1 𝑥 3 4) 𝑦= 𝑥 2 +3

Integration L.O. All pupils understand what integration is All pupils can integrate individual terms

This is known as integrating! Main 1: what integration is If we wanted to differentiate we would do the following: xn xn multiply by the power reduce the power by 1 nxn-1 Integration is the opposite of differentiation. divide by the power increase the power by 1 xn This is known as integrating!

This is known as integrating! Main 1: what integration is Notation: 𝑥 2 𝑑𝑥 This is the symbol we use to show that we are integrating. It’s like a stretched out S We are integrating the function x2 with respect to x divide by the power increase the power by 1 xn This is known as integrating!

Main 1: We need to integrate to find y! 𝑑𝑦 𝑑𝑥 = 𝑥 2 what integration is Example 1 𝑑𝑦 𝑑𝑥 = 𝑥 2 Integration is the opposite of differentiation, so it allows us to find what y was originally equal to. We need to integrate to find y!

Main 1: what integration is Example 1 𝑥 2 𝑑𝑥 Lets check! 𝑑𝑦 𝑑𝑥 = 𝑥 2

Main 1: 𝑦= 𝑥 3 3 +2 what integration is However our original equation could have been 𝑦= 𝑥 3 3 +2 If we differentiate we get But if we integrate that we get Which isn’t our original equation!

𝑥 2 𝑑𝑥 = 𝑥 3 3 +𝑐 Main 1: what integration is So we have to write a general solution, 𝑥 2 𝑑𝑥 = 𝑥 3 3 +𝑐 This ‘plus c’ generalises our integration, next lesson we will find c!

Main 1: what integration is Example 2 3𝑥 𝑑𝑥

Main 1: what integration is Example 2 8 𝑥 3 + 1 𝑥 2 +3 𝑑𝑥

Main 1: what integration is Have a go in pairs 6 𝑥 5 + 1 𝑥 3 −4𝑥 𝑑𝑥

Main 1: what integration is Example 3 𝑡 2 + 𝑡 𝑑𝑡

Integration L.O. All pupils understand what integration is All pupils can integrate individual terms

integrate individual terms Main 2: integrate individual terms Find these integrals 5) 𝑥 −4 𝑑𝑥 1) 3 𝑥 2 𝑑𝑥 6) 3 𝑥 2 +2 𝑥 𝑑𝑥 2) 3𝑥 𝑑𝑥 3) 5 𝑥 4 +2 𝑑𝑥 7) ( 𝑥 2 −2) 2 𝑑𝑥 8) 𝑥 2 3 𝑑𝑥 4) 𝑥 −2 𝑑𝑥

integrate individual terms Main 2: integrate individual terms Answers 5)− 𝑥 −3 3 1) 𝑥 3 2) 3 2 𝑥 2 6) 𝑥 3 + 4 3 𝑥 3 2 7) 𝑥 5 5 − 4𝑥 3 3 +4𝑥 3) 𝑥 5 +2𝑥 4)− 𝑥 −1 8) 3 5 𝑥 5 3

integrate individual terms Main 2: integrate individual terms Example 4 Find y in terms of x if 𝑑𝑦 𝑑𝑥 = 4 𝑥 6 −𝑥 𝑥 3

integrate individual terms Main 2: integrate individual terms Example 5 Find 𝑦 𝑑𝑥 given that 𝑦= 𝑥 2 +2𝑥 𝑥

Integration L.O. All pupils understand what integration is All pupils can integrate individual terms