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The Proof is Trivial! Watch

1. (Original post by bogstandardname)
Problem 11* (if you have never seen substitution before, **)

Find the general solution, by a suitable substitution or otherwise of this differential equation.

Solution 12 (a bit rough-and-ready, I'm afraid, and I feel like there should be a nice "otherwise"):
Spoiler:
Show

Since the denominator involves cos x, I'll use the simplest possible substitution I can think of that will interact nicely with the cos x on the denominator:
Substitute , so that .
Then:
.
Hence:
, and so .
This is separable (I'll take out another fraction 1/2, to make the reverse-chain-rule more obvious):
, and so and . Substituting back for y:
.
We can solve this for y, to get:
.
2. (Original post by Lord of the Flies)
Problem 15*/**

Evaluate
Hmm not totally sure how to do this (although I have an idea)

If you let then for all . I believe the fact it doesn't converge to 0 at one point isn't a problem with this (Lebesgue)

So by Lebesgue

Disclaimer: There's a very good chance this is wrong (I haven't covered Lebesgue )
3. Problem 16 **/***

Let be a function such that .

Define and .

Show that .

Find .
4. (Original post by Noble.)
Hmm not totally sure how to do this (although I have an idea)

If you let then for all . I believe the fact it doesn't converge to 0 at one point isn't a problem with this (Lebesgue)

So by Lebesgue

Disclaimer: There's a very good chance this is wrong (I haven't covered Lebesgue )
Spoiler:
Show
The fact that it doesn't converge to 0 at one is in fact a problem. Example:

Even though
5. I'm getting these from somewhere but I think some of these are beautiful problems which hopefully someone with C1-4 can do (except 18, which ironically might be the easiest). Hope you don't mind the influx!

Problem 17*

Let be an arithmetic sequence and be a geometric sequence. The first four terms of is 0, 0, 1 and 0.

Find .

Problem 18**

Evaluate

Problem 19*

If , evaluate

Problems 20 and 21 are below. I told you I'd spam this thread

Problem 22*

Evaluate

Problem 23*

Let denote the set of triples such that .

Evaluate

(I wanted to edit this to restrict the choice of n, but actually it makes little difference. Do people need a hint?)
6. Problem 20 **

By using a suitable substitution, or otherwise, show that

Show further that the area of a circle, A, satisfies

7. Problem 21 - */**

The uniqueness theorem of anti-derivatives states that, if , then .

By considering the derivatives of and , verify that .

By considering suitable derivatives, prove the following identities:

i)

ii)

iii) .

Deduce that the results in parts i) and ii) hold independently of the base of the logarithm.

Spoiler:
Show

This was a question I came up with when there was talk about a user-contributed STEP paper being made by TSR members. However, I think here is a better place for it.

8. Solution 17

Let and . Then:

Solving simultaneously, we obtain , , and .

Hence .
9. Solution 19

.
10. (Original post by und)
Solution 17

Let and . Then:

Solving simultaneously, we obtain , , and .

Hence .

Right, where have I gone wrong??
I didn't say the answer was pretty! But I meant s_10
11. (Original post by shamika)

Problem 18**

Evaluate
Solution 18
Spoiler:
Show
Since as x becomes large, we have which clearly goes to infinity, being a product of two things which go to infinity. (since goes to infinity, so does x to the power of that.)
12. (Original post by Smaug123)
Solution 18
Spoiler:
Show
Since as x becomes large, we have which clearly goes to infinity, being a product of two things which go to infinity. (since goes to infinity, so does x to the power of that.)
I really need to stop posting when I'm exhausted - I've corrected the problem, sorry.
13. (Original post by Lord of the Flies)
Problem 15*/**

Evaluate

What if the upper limit in the integral were fixed, say

Hi

Take the binomial expansion of n/(1+x^2)^-1 (far too tedious to write out all the working!)
Then simplify and integrate, eliminate the n terms, and you're left with 1 - 1/2 + 1/3 - 1/4... which is of course the taylor expansion of ln(2)

sorry I don't know how to use LaTex so it would be a lot of effort to type out the working, but it should be straightforward to follow the steps
14. Solution 22

Let . Then by parts, . Hence
15. (Original post by und)
Solution 19

.
yep!
16. (Original post by shamika)
I really need to stop posting when I'm exhausted - I've corrected the problem, sorry.
Heh, I did wonder; you did say it might be the easiest :P

Problem 18**

Evaluate

Solution 18
Spoiler:
Show
by considering and using the continuity of log, and the fact that .
Therefore, .
17. (Original post by und)
Solution 22

Let . Then by parts, . Hence
Yep!

(Original post by Smaug123)
Heh, I did wonder; you did say it might be the easiest :P

Problem 18**

Evaluate

Solution 18
Spoiler:
Show
by considering and using the continuity of log, and the fact that .
Therefore, .
Yep!
18. (Original post by und)
Solution 17

Let and . Then:

Solving simultaneously, we obtain , , and .

Hence .
Yep. This isn't as nice as I'd tried to make it, but oh well, it was meant to entice some lurkers to solve something which is essentially C1/2 standard
19. (Original post by _Izzy)
...
Well done I'll type it out for you (credit to Izzy in the OP):

Hence the limit is

Replacing with obviously makes no difference.
20. Solution 21

Let and . Then . Letting gives as required.

i) Let and . Then . Letting gives as required.

ii) Let and . Then . Letting gives as required.

iii) Let and . Then . Letting gives as required.

Any logarithm can be mapped onto the natural logarithm using a linear transformation, hence the results still hold.

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Updated: December 11, 2017
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