“Teach A Level Maths” Vol. 2: A2 Core Modules 28: Integration giving Logs © Christine Crisp

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When we differentiated we found the general result with its derivative “on top” The inner function . . .

When we differentiated we found the general result This result came from using the chain rule. e.g. Differentiate with respect to x. Solution: Let

Since indefinite integration is the reverse of differentiation

Since indefinite integration is the reverse of differentiation The modulus signs are there because the log of a negative number is not defined.

e.g. 1

e.g. 1

e.g. 1

e.g. 1

e.g. 1 e.g. 2 We don’t need the mod signs since is always positive. e.g. 3

Some integrals can be adjusted so that they lead to logs. We need 2x in the numerator . . . so we write 2x We have multiplied by 2, so to compensate, we must divide by 2.

Some integrals can be adjusted so that they lead to logs. Now the can just stay where it is, multiplying the integral.

e.g. 5 What do we want in the numerator? Ans: We now need to get rid of the minus and replace the 3. . . .

e.g. 5 What do we want in the numerator? Ans: We now need to get rid of the minus and replace the 3. Always check by multiplying the numerator by the constant outside the integral:

e.g. 5 What do we want in the numerator? Ans: We now need to get rid of the minus and replace the 3.

e.g. 6 . . .

e.g. 6

BUT we can never adjust x e.g. is not

a multiplying constant SUMMARY Quotients consisting of a function with its derivative “on top” can be integrated directly. The integral is the natural log of . Quotients can be adjusted only if a multiplying constant needs to be changed.

Exercises Integrate the following: 1. 2. 3. 4. 5. 6. ( I haven’t made a mistake here ! )

Solutions: 1. 2. Mod signs here aren’t wrong but are unnecessary. 3. 4.

Solutions: 5. 6. Using the 3rd log law we can write this as giving

Definite integration There is no difference in method but we are expected to give exact answers and to simplify the logs. e.g. 1 The curved brackets are essential

e.g. 2 We have to be very careful with this minus sign The curved brackets are essential

Exercises Evaluate the following: 1. 2.

1. Solutions:

Solutions: 2.

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SUMMARY Quotients consisting of a function with its derivative “on top” can be integrated directly. The integral is the natural log of . Quotients can be adjusted if only a multiplying constant needs to be changed.

e.g. 1 e.g. 2 e.g. 3

e.g. 4 e.g. 5 e.g. 6

Definite integration There is no difference in method but we are expected to give exact answers and to simplify the logs. e.g. 1 The curved brackets are essential

e.g. 2 We have to be very careful with this minus sign