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The hard integral thread.

Had fun discussing integrals with you lot yesterday. Lets all post some more and restrict the discussion to this thread.

I'll start

In=01(1x)ndx I_n = \displaystyle\int_0^1 (1-\sqrt{x})^n \,dx

Find a reduction formula.

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Original post by Louisb19
Had fun discussing integrals with you lot yesterday. Lets all post some more and restrict the discussion to this thread.

I'll start

In=01(1x)ndx I_n = \displaystyle\int_0^1 (1-\sqrt{x})^n \,dx

Find a reduction formula.


I had to cheat, as it's easy enough to integrate directlym via a substitution. Having found the integral, I then construct a reduction formula.

Spoiler

(edited 8 years ago)
Reply 2
Original post by ghostwalker
I had to cheat, as it's easy enough to integrate directlym via a substitution. Having found the integral, I then construct a reduction formula.

Spoiler



Feel free to contribute your own integral question if you have one. I'm bored :frown:
Original post by Louisb19
Feel free to contribute your own integral question if you have one. I'm bored :frown:
We've had the 0π/2logsinxdx\displaystyle \int_0^{\pi / 2} \log \sin x \, dx.

In a similar vein (and easier IMHO):

0π/4log(1+tanx)dx\displaystyle \int_0^{\pi / 4} \log (1+ \tan x) \, dx.
Also, if you want to feel inadequate, have a look at: http://math.stackexchange.com/users/97378/cleo
Original post by DFranklin
Also, if you want to feel inadequate, have a look at: http://math.stackexchange.com/users/97378/cleo


I feel out of depth with the level of some people on here. Then I go on SE and wonder why I even bother.
I feel like these integrals would be best suited to: http://www.thestudentroom.co.uk/showthread.php?t=2313384
I posted this before (can't remember if it was on this site or elsewhere) either way if you haven't seen it before it is quite a cool one.

xn1+x+x22+...+xnn! dx\displaystyle \int \frac{x^n}{1+x+\frac{x^2}{2}+...+\frac{x^n}{n!}}~dx .
Reply 8
Original post by DFranklin
We've had the 0π/2logsinxdx\displaystyle \int_0^{\pi / 2} \log \sin x \, dx.

In a similar vein (and easier IMHO):

0π/4log(1+tanx)dx\displaystyle \int_0^{\pi / 4} \log (1+ \tan x) \, dx.


I think I've got the first one.


The question is ln(sinx) \ln(sinx) right?
(edited 8 years ago)
Reply 9
Subbing. Let's integrate :sexface:

From one of the trinity pre interview tests: integrate 1/(x + root(1-x^2))

Posted from TSR Mobile
Reply 10
integral easy lol.png
that was my FP2 prep a few days ago lol
Reply 11
Original post by poorform
I posted this before (can't remember if it was on this site or elsewhere) either way if you haven't seen it before it is quite a cool one.

xn1+x+x22+...+xnn! dx\displaystyle \int \frac{x^n}{1+x+\frac{x^2}{2}+...+\frac{x^n}{n!}}~dx .


I feel like there should be a nicer expression but I'm tired and broken by assignments

Spoiler

Integrate [3cosxsin(^2)x]
A little messy, but ya'll might like it:
11+x4 dx\displaystyle\int \dfrac{1}{1+x^4} \ dx.

Original post by Louisb19
I think I've got the first one.


The question is ln(sinx) \ln(sinx) right?


The two are often used interchangeably beyond A-Level, though to be fair, even if it were log10\log_{10} or for that matter any other base it's just a constant scaling.
(edited 8 years ago)
Here's an easy one, which is more theory than fancy tricks:

Let I=π216π24sinxx dx\displaystyle I = \int_{\frac{\pi^2}{16}}^{\frac{ \pi ^2}{4}} \frac{\sin \sqrt{x}}{x} \ dx

Show that 2ln2I2ln2\sqrt{2} \ln 2 \le I \le 2 \ln 2
Reply 15
Original post by DFranklin
We've had the 0π/2logsinxdx\displaystyle \int_0^{\pi / 2} \log \sin x \, dx.

In a similar vein (and easier IMHO):

0π/4log(1+tanx)dx\displaystyle \int_0^{\pi / 4} \log (1+ \tan x) \, dx.


Ahh I don't have this one, care to give me a hint?
Original post by Mihael_Keehl
Integrate [3cosxsin(^2)x]


Was that in an FP2 paper.. June 2015 perhaps? :K:

By recognition - sin^3(x)
Reply 17
Original post by SeanFM
Was that in an FP2 paper.. June 2015 perhaps? :K:

By recognition - sin^3(x)


Pretty sure it was. Bit harsh that that paper was 71 for an A* lol
Original post by joostan
A little messy, but ya'll might like it:
11+x4 dx\displaystyle\int \dfrac{1}{1+x^4} \ dx.


Spoiler




The two are often used interchangeably beyond A-Level, though to be fair, even if it were log10\log_{10} or for that matter any other base it's just a constant scaling.


log is used in confusingly different ways between mathematics, engineering and physics to mean both log_10 and log_e. It's a bit of a mess.
Reply 19
Original post by 1 8 13 20 42
Pretty sure it was. Bit harsh that that paper was 71 for an A* lol


I did it a while ago and it seemed to be one of the easiest FP2 papers to date.

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