Hi, I'm Shaun! I'm currently studying for a masters in Mathematics at the University of Exeter.
I believe that it is very important in mathematics to understand what you are taught, not just blindly learn formulae! As such, I focus on teaching the ideas behind the method, not just the method itself.
I also think it's very important to practise questions yourself, so the sessions are interactive.
The actual structure of the sessions is up to you! I can go through exam papers or focus on particular topics you struggle with.
If you have any questions, feel free to contact me. You can also arrange a free 'Meet the Tutor' session through this website to get to know me a little before you decide whether to go ahead with some sessions.
|Maths||A Level||£20 /hr|
|Before 12pm||12pm - 5pm||After 5pm|
Please get in touch for more detailed availability
Find the following integral: ∫ x sin(x) dx
This question is a good candidate for the integration by parts method, as it is the product of two different 'parts'.
Recall that if you have an integral of the form
∫ u(dv/dx) dx
it can be written as
uv – ∫ v(du/dx) dx.
We need to decide which part we will differentiate (as in, which part is u), and which part we will integrate (as in, which part is dv/dx).
We can note that continuously differentiating sin(x) results in a loop of cos(x), –sin(x), –cos(x), sin(x)..., whereas differentiating x once gives 1.
From this, it seems to make sense that we would want to differentiate the x part (so u is x) and therefore integrate the sin(x) part (so dv/dx is sin(x) ). So, let
u = x, which implies du/dx = 1
dv/dx = sin(x). Integrating this to get v gives v = –cos(x).
So our integral is now of the form required for integration by parts.
∫ x sin(x) dx
= ∫ u(dv/dx) dx
= uv – ∫ v(du/dx) dx
= –x cos(x) – ∫ –cos(x)*1 dx
= –x cos(x) – ∫ –cos(x) dx
= –x cos(x) + ∫ cos(x) dx
The integral of cos(x) is equal to sin(x). We can check this by differentiating sin(x), which does indeed give cos(x). Finally, as with all integration without limits, there must be a constant added, which I'll call c. So the final answer is
∫ x sin(x) dx = –x cos(x) + sin(x) + csee more