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# Step ii 2007 q6

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1. I'm not sure if math questions for STEP are meant to be posted in separate posts. So I apologise if I haven't gotten it right. Feel free to merge this post.

Actual question:

.
.

http://www.thestudentroom.co.uk/atta...4&d=1183161671

First Question --- How would you see this trick --- to add to the desired integral?

I know the question asks us to differentiate the functions but I still don't see this trick.

Then use to get

Second Question --- I know how to integrate by using the substitution . But the STEP solution says that no calculus work is needed for this part --- the integration of . I don't see the derivatives in the first part help me to get ?

Thanks a lot---
I'm not sure if math questions for STEP are meant to be posted in separate posts. So I apologise if I haven't gotten it right. Feel free to merge this post.

Actual question:

.
.

http://www.thestudentroom.co.uk/atta...4&d=1183161671

First Question --- How would you see this trick --- to add to the desired integral?

I know the question asks us to differentiate the functions but I still don't see this trick.

Then use to get

He seems to have used a less simplified form of the derivative of . Namely, by the product rule, . As soon as he saw that, it's obvious that he'll be able to evaluate an integral similar to that so he went for the addition-subtraction trick.

Second Question --- I know how to integrate by using the substitution . But the STEP solution says that no calculus work is needed for this part --- the integration of . I don't see the derivatives in the first part help me to get ?

Thanks a lot---
He then noted a different form of the above derivative: (1)

and also:
(2)

Furthermore:

by (1) and (2), which is easy to integrate. He made a mistake in the last line of working - There's a small sign error.

I have to say, I think this is the long way of doing it. When I did the question, I noted that:

Using (2), the RHS is easy to integrate and you're done.
3. Great! Thanks ---
4. (Original post by Farhan.Hanif93)
I have to say, I think this is the long way of doing it. When I did the question, I noted that:

Using (2), the RHS is easy to integrate and you're done.
actually, one last question ---

How did you see the above? It wasn't and is still not obvious to me. The only way I see it is ---

and it looks like only to be a very clever algebra trick ---

Thanks again ---
actually, one last question ---

How did you see the above? It wasn't and is still not obvious to me. The only way I see it is ---

and it looks like only to be a very clever algebra trick ---

Thanks again ---
I think you've made a small typo at the start; it should be . My reasoning behind it was that I wanted to rewrite this derivative in terms of AND a constant multiple of the same or other derivative. It was clear to me that can be rewritten in that form so I did the working and it just fell out.
6. How I did it was as follows:

and

Now we can easily see that
7. Thank you very much ---

I've also fixed my typo.

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