Integration mega collection Watch

TeeEm
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I am attaching in this thread an integration booklet with 622 indefinite integrals at present ( 5 new ones to be added shortly)

This is the student version of the booklet which has no answers or solutions.
(A carbon copy of this document with answers and full solutions also exists but it is too large to attach here)

Can you suggest any decent "beautifully disgusting" integrals to add to this collection.
Indefinite integrals only please.
(undergrad techniques are ok, within reasonable difficulty)


integration_indefinite_mixed_up_student_version_condense.pdf


The booklet is not in strict "hardness" order but overall the questions get progressively harder

Many thanks

Note that at Q350 onwards a large proportion of integrals will be only accessible to further mathematicians
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Hasufel
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Kickass!
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Notnek
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(Original post by TeeEm)
I am attaching in this thread an integration booklet with 622 indefinite integrals at present ( 2 new ones to be added shortly)

This is the student version of the booklet which has no answers or solutions.
(A carbon copy of this document with answers and full solutions also exists but it is too large to attach here)

Can you suggest any decent "beautifully disgusting" integrals to add to this collection.
Indefinite integrals only please.
(undergrad techniques are ok, within reasonable difficulty)


integration_indefinite_mixed_up_student_version_condense.pdf


The booklet is not in strict "hardness" order but overall the questions get progressively harder

Many thanks

Note that at Q350 onwards a large proportion of integrals will be only accessible to further mathematicians

The booklet is not in strict "hardness" order but overall the questions get progressively harder
This should be the new FP3 exam : attempt as many of the 622 integrals as you can before the time runs out / your brain explodes.

But seriously, this is an extremely useful set of questions - one of the best further maths resources I've see in a while.
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TeeEm
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(Original post by notnek)
This should be the new FP3 exam : attempt as many of the 622 integrals as you can before the time runs out / your brain explodes.

But seriously, this is an extremely useful set of questions - one of the best further maths resources I've see in a while.
This is the "mother of all integration booklets" (I started the collection around 2004)

Shame I cannot upload the main version with answers and solutions.

I hope further contributions can be made because I have run out of ideas.

BTW this is not further maths to start with. My straight mathematicians usually do the first 100 to 150 as part of their A2 preparation
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Notnek
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My suggestion for a disgusting integral:

\displaystyle \int \frac{1}{(\cos^2 x +4\sin x - 5)\cos x} \ dx

Probably more because of it's length as opposed to its difficulty.
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TeeEm
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(Original post by notnek)
My suggestion for a disgusting integral:

\displaystyle \int \frac{1}{(\cos^2 x +4\sin x - 5)\cos x} \ dx

Probably more because of it's length as opposed to its difficulty.
If it is not there it will be added

thanks


EDIT: Just did it. I must remember to rate this contribution because it does not let me.
It Is hard enough without any hints
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Hasufel
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this one looks simple (i dare ya!):

\displaystyle \int \frac{\sin(x)}{1-\tan(x)}dx

(equivalent to \displaystyle \int \frac{\sin(x)\cos(x)}{\cos(x)-\sin(x)}dx )
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TeeEm
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(Original post by Hasufel)
this one looks simple (i dare ya!):

\displaystyle \int \frac{\sin(x)}{1-\tan(x)}dx

(equivalent to \displaystyle \int \frac{\sin(x)\cos(x)}{\cos(x)-\sin(x)}dx )

definitely to be tried ... If it is in the list it will be added.

Sorry I cannot rep you cause I did rep today your contribution into a limits question.

Thanks
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Hasufel
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(Original post by TeeEm)
definitely to be tried ... If it is in the list it will be added.

Sorry I cannot rep you cause I did rep today your contribution into a limits question.

Thanks
it`s an absolute ***** to do! - wouldn`t think it by looking, though!
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TeeEm
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(Original post by Hasufel)
it`s an absolute ***** to do! - wouldn`t think it by looking, though!
I have gone so far as -1/2[cosx - sinx - 1/(cosx-sinx)] which I think can be done ...


My brain has stalled at 1/(cosx -sinx) and can only think little t at present ...

definitely will be added if not there
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tombayes
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\int cos^n(x)dx, n \in \mathbb{N}
\int sin^n(x)dx, n \in \mathbb{N}
\int tan^n(x)dx, n \in \mathbb{N}

Also, wolfram alpha will give you answers in terms of hyper-geometric functions but you do not need them if you consider the different cases resulting from different choices of n.

(Original post by TeeEm)

The booklet is not in strict "hardness" order but overall the questions get progressively harder

Good old 622: \int \sqrt{tan(x)} dx that was on one of my first (undergrad) uni problem sheets
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TeeEm
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(Original post by tombayes)
\int cos^n(x)dx, n \in \mathbb{N}
\int sin^n(x)dx, n \in \mathbb{N}
\int tan^n(x)dx, n \in \mathbb{N}

Also, wolfram alpha will give you answers in terms of hyper-geometric functions but you do not need them if you consider the different cases resulting from different choices of n.
up to n=4 i have these I think

after 5 you really need reduction formulas for indefinite integrals, I know no other technique
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tombayes
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(Original post by TeeEm)
up to n=4 i have these I think

after 5 you really need reduction formulas for indefinite integrals, I know no other technique
hint: binomial expansion but need to write cos, sin, or tan in a different form...
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TeeEm
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(Original post by tombayes)
hint: binomial expansion but need to write cos, sin, or tan in a different form...
... very clever ...





EDIT: owe you a rep I hope I remember it
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Hasufel
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(Original post by tombayes)
hint: binomial expansion but need to write cos, sin, or tan in a different form...
you mean the \displaystyle cos(x)=\frac{1}{2}(z+\frac{1}{z}  ) etc thingy...?

(i don`t know how to make brackets bigger in LaTex)
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Slowbro93
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*get's my pen and paper*
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tombayes
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(Original post by Hasufel)
you mean the \displaystyle cos(x)=\frac{1}{2}(z+\frac{1}{z}  ) etc thingy...?

(i don`t know how to make brackets bigger in LaTex)

I was actually thinking of

cos(x)=\frac{e^{ix}+e^{-ix}}{2} but yours is essentially the same.
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tombayes
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(Original post by TeeEm)
... very clever ...

Try this one:

\int e^{\alpha x}[\ cos(\beta x)+ \sin(\gamma x)] dx
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TeeEm
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(Original post by tombayes)
Try this one:

\int e^{\alpha x}[\ cos(\beta x)+ \sin(\gamma x)] dx
I have that one (by complex or parts), but I will add some powers of sin/cos/tan via complex route as you suggested
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TeeEm
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(Original post by tombayes)
\int cos^n(x)dx, n \in \mathbb{N}
\int sin^n(x)dx, n \in \mathbb{N}
\int tan^n(x)dx, n \in \mathbb{N}

Also, wolfram alpha will give you answers in terms of hyper-geometric functions but you do not need them if you consider the different cases resulting from different choices of n.




Good old 622: \int \sqrt{tan(x)} dx that was on one of my first (undergrad) uni problem sheets
LOL!!!

622 was in my undergrad sheet too!! As a tribute I always left it last!

I went UCL. You?
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