I don't understand integration by parts - can you explain it please?

(Apologies I don't have the relevant diagrams here so this reads longer than it actually is on paper)

1) First it's important to establish that the prior knowledge is there before explaining something more advanced.

"Sure, so do you understand how to evaluate a standard integral? (give example of "standard integration" question)

"So we can solve this either straight away with the usual method, or by using the u-sub method to simplify the integral (again check they're familiar with both of these as it's crucial to understand before moving on)

Once the required knowledge is there we can answer the question.

2) Firstly, explain the new concepts - "what's different, why is it different, and what do I need to know to tackle it?"

"However, sometimes you'll be asked integration questions like this (example of "integration by parts" question) so can you see the difference here? (encourage possible explanations, gives an indication as to the student's affinity for learning new concepts)

Yes/It's actually that we have two functions multiplied together which means they cannot be solved with the previous method.

Instead we use this formula to solve it: integral(udv) = uv - integral(vdu) (explain whether this is given in the exam or if it needs to be memorised)

3) Next, work through an example question slowly, explaining each step - "how do I use what I know to solve a question?"

"We have integral (xsinx), so by setting u=x and dv/dx=sinx, we obtain du/dx = 1 and v = -cosx

Then by using the equation and plugging in these values, we get to integral(xsinx) = -xcosx - integral(-cosx*1)

So we can now integrate this new integral, thus integral(xsinx) = -xcosx - (-sinx) + c i.e. equal to sinx - xcosx +c"

Here we have worked through an example question step-by-step, making sure the student follows along. Make sure this is clear before moving on to more complicated examples.

4) Next, explain a few more of the different types of questions that will arise.

"The example question you did before used a polynomial and a trigonometric function, which is one type of question you can get, but another is an integral with two trigonometrics..." then provide example question and walk through answer again, highlighting the key differences.

5) Finally, make a note to self to include these types of questions in homework for the student.

Time can be short in tutorials and it's easy to spend the majority of the time available on one/two concepts, which may not always be the best use of time. Ensure that the key concepts and ideas are explained and examples provided, and allow the student to practice it in their own time and follow up with more detailed questions later.