The hard integral thread. Watch

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

I'll start

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

Find a reduction formula.
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ghostwalker
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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

 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:
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I_n=\frac{n}{n+2}I_{n-1}

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Louisb19
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(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:
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I_n=\frac{n}{n+2}I_{n-1}

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

In a similar vein (and easier IMHO):

\displaystyle \int_0^{\pi / 4} \log (1+ \tan  x) \, dx.
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DFranklin
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Also, if you want to feel inadequate, have a look at: http://math.stackexchange.com/users/97378/cleo
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poorform
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(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.
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rayquaza17
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I feel like these integrals would be best suited to: http://www.thestudentroom.co.uk/show....php?t=2313384
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poorform
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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.

\displaystyle \int \frac{x^n}{1+x+\frac{x^2}{2}+...  +\frac{x^n}{n!}}~dx .
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Louisb19
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(Original post by DFranklin)
We've had the \displaystyle \int_0^{\pi / 2} \log \sin x \, dx.

In a similar vein (and easier IMHO):

\displaystyle \int_0^{\pi / 4} \log (1+ \tan  x) \, dx.
I think I've got the first one.


The question is  \ln(sinx) right?
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Krollo
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Subbing. Let's integrate :sexface:

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

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Dabo_26
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Name:  integral easy lol.png
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that was my FP2 prep a few days ago lol
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math42
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(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.

\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
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Is it n!(x - ln(the denominator in that integral))
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Mihael_Keehl
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Integrate [3cosxsin(^2)x]
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joostan
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A little messy, but ya'll might like it:
\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) right?
The two are often used interchangeably beyond A-Level, though to be fair, even if it were \log_{10} or for that matter any other base it's just a constant scaling.
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atsruser
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Here's an easy one, which is more theory than fancy tricks:

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

Show that \sqrt{2} \ln 2 \le I \le 2 \ln 2
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Louisb19
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(Original post by DFranklin)
We've had the \displaystyle \int_0^{\pi / 2} \log \sin x \, dx.

In a similar vein (and easier IMHO):

\displaystyle \int_0^{\pi / 4} \log (1+ \tan  x) \, dx.
Ahh I don't have this one, care to give me a hint?
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Kevin De Bruyne
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(Original post by Mihael_Keehl)
Integrate [3cosxsin(^2)x]
Was that in an FP2 paper.. June 2015 perhaps?

By recognition - sin^3(x)
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math42
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(Original post by SeanFM)
Was that in an FP2 paper.. June 2015 perhaps?

By recognition - sin^3(x)
Pretty sure it was. Bit harsh that that paper was 71 for an A* lol
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atsruser
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(Original post by joostan)
A little messy, but ya'll might like it:
\displaystyle\int \dfrac{1}{1+x^4} \ dx.
Spoiler:
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I'm too tired to do all of the work, but I wrote:

x^4+1=(x^2+ax+1)(x^2+bx+1)

which after some easy algebra gives:

x^4+1=(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)

and from thence it's partial fractions, I guess.


The two are often used interchangeably beyond A-Level, though to be fair, even if it were \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.
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Louisb19
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(Original post by 13 1 20 8 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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