The Proof is Trivial!

Problem 178*

Evaluate $\displaystyle\int_0^2 x^5 \sqrt{1-x^3} dx$

Solution 174

Let $x = \tan \theta :$

$\displaystyle\int_{0}^{\infty} \dfrac{\ln x}{1 + x^{2}} dx = \displaystyle\int_{0}^{ \frac{\pi}{2}} \ln \left( \tan \theta \right) d \theta$

$= \displaystyle\int_{0}^{ \frac{\pi}{2}} \ln \left( \sin \theta \right ) d \theta - \displaystyle\int_{0}^{ \frac{\pi}{2}} \ln \left( \cos \theta \right) d \theta = 0$

Are you sure the upper limit is supposed to be 2? Otherwise the integral is not defined for real numbers past 1...

In light of it being a "speed" competition, allow me to direct you toward my solution

Solution 174 (2)

Let $x \to x^{-1}$,

$\displaystyle \Rightarrow \int_0^{\infty} \frac{\ln(x)}{1+x^2} dx = \int_0^{\infty} -\frac{\ln(x)}{1+x^2} dx =0 \ \square$

Felix you need to relable your questions, there already is a 171

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Problem 185*

Find all $f:\;\mathbb{R}\to\mathbb{R}$ such that:



Problem 186*

A nice problem. Prove/disprove that there exists $f:\;\mathbb{N}\to\mathbb{N}$ such that:



$f^k$ means $f$ applied $k$ times in this case.

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Well, to those who were unhappy about the lack of (*)'s, you now have loads. Go go go go!

It takes like 10 seconds less Neat, nonetheless

Meh

Please post some interesting series and/or integration questions

Lol jk functional equations are not * questions

Yes they are. In fact a lot of functional equations only require GCSE knowledge.

But the concept of a function isn't introduced until a-level

But the concept of a function isn't introduced until a-level

Problem 178*

Evaluate $\displaystyle\int_0^2 x^5 \sqrt{1+x^3} dx$

I'm pretty sure I've seen function-related GCSE questions on the maths forum but whatever.

Let $x^3 = sinh^2u$

$3x^2 \ dx = 2sinhucoshu \ du$

$\Rightarrow I = \frac{2}{3} \int sinh^3ucosh^2u \ dx$

$= \frac{2}{3} \int sinhu(cosh^2u-1)cosh^2u \ dx$

$= \frac{2}{3} \int sinhucosh^4u - sinhucosh^2u \ dx$

$= = \frac{2}{3} [ \frac{1}{5}cosh^5u -\frac{1}{3}cosh^3u ]$

x = 0 so u = 0, coshu = 1, x = 2 so coshu = 3

Finally plugging those in I get 1192/45, although this isn't really a nice answer...

Edit: I thought the first integral you posted was $\displaystyle\int_{0}^{\infty} \dfrac{ \ln x}{(x^{2} + 1)^{2}} \ dx$ and I thought no way in hell can anyone do that in 20 minutes

Or u =1+x^3 ?