# Teacher Notes

### Why use this resource?

How do we integrate $\frac1x$? This resource offers an approach via transforming the graph of $y=\frac1x$ without needing calculus tools. It is a structured resource designed to lead students to the amazing result that the area under the graph behaves as a logarithm. Many students have been intrigued to see how seemingly disparate topics (transformations, integration and logs) combine to give this result.

### Preparation

Students will need a copy of the diagrams page to write on. It may be helpful to print the problem page too so that students can easily work through the questions.

### Possible approach

The Warm-up section reviews the effect on an equation of stretching its graph. It uses an example similar to $y=\frac1x$, but where the behaviours of stretches in the $x$ and $y$ directions differ from each other. This will help to clarify whether students understand the algebra involved.

Asking initial questions such as “What is the integral of $x^n$? Does this rule work for $x^{-1}$?” can help to set the scene for the main problem.

The idea of using a function to represent an integral – here, $I(a)=\int_1^a \frac{1}{x}\,dx$ – might be unfamiliar to students, and may need some further explanation.

### Key questions

• How are $I(a)$ and $I(b)$ related?

• What does your answer suggest about the function $I(x)$?

• Once you know the value of $a$ which gives $I(a)=1$, what does this tell you?

### Possible extension

The “Some further questions” part can be used for further discussion of closely related ideas.

Another interesting question to ask is “What is special about the function $y=\frac1x$ which makes this technique work?”