The Proof is Trivial!

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A nice little continuation with the theme from IrrationalRoot's problem.

Problem 615*

Evaluate

\displaystyle \int\sqrt{x+\sqrt{x+\sqrt{x+...}}} \mathrm{d}x

Still procrastinating so solution:

Spoiler

Reply 3881

123Master321

This one's nice: Problem 617*:

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

Reply 3882

username3555092

Original post by 123Master321

This one's nice: Problem 617*:

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

Unparseable latex formula:

\displaystyle [br][br]\begin{align*}\int \frac{4x^5-1}{(x^5+x+1)^2} \mathrm{d}x &= \int \frac{(-1)(x^5+x+1)-(-x)(5x^4+1)}{(x^5+x+1)^2} \mathrm{d}x\\ &= \frac{-x}{x^5+x+1}+C \end{align*}

Did take a bit of luck in trying the quotient rule and fiddling with

P'(x)(x^5+x+1)-P(x)(5x^4+1) = 4x^5-1

, was surprised it worked out so nicely.

(edited 6 years ago)

Reply 3883

123Master321

Original post by I hate maths

Unparseable latex formula:

\displaystyle [br][br]\begin{align*}\int \frac{4x^5-1}{(x^5+x+1)^2} \mathrm{d}x &= \int \frac{(-1)(x^5+x+1)-(-x)(5x^4+1)}{(x^5+x+1)^2} \mathrm{d}x\\ &= \frac{-x}{x^5+x+1}+C \end{align*}

Did take a bit of luck in trying the quotient rule and fiddling with

P'(x)(x^5+x+1)-P(x)(5x^4+1) = 4x^5-1

, was surprised it worked out so nicely.

Thats was the method that I found. Well done!

Reply 3884

BluedRose

I know this thread hasn't been posted in for around a year, but I'd like to share that I've parsed the first 500 or so questions from this thread. You can find a pdf here (and also the script used if you wanna parse yourself).

It's not perfect but it seems good enough for anyone still poking around this thread from time to time :smile: