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Puzzles

An integral

Source: Alex Bolton (inspired by Book Proofs blog)
What is
$$\int_0^{\frac\pi2}\frac1{1+\tan^a(x)}\,dx?$$

Find them all

Find all continuous positive functions, $$f$$ on $$[0,1]$$ such that:
$$\int_0^1 f(x) dx=1\\ \mathrm{and }\int_0^1 xf(x) dx=\alpha\\ \mathrm{and }\int_0^1 x^2f(x) dx=\alpha^2$$

Integrals

$$\int_0^1 1 dx = 1$$
Find $$a_1$$ such that:
$$\int_0^{a_1} x dx = 1$$
Find $$a_2$$ such that:
$$\int_0^{a_2} x^2 dx = 1$$
Find $$a_n$$ such that (for $$n>0$$):
$$\int_0^{a_n} x^n dx = 1$$

Double derivative

What is
$$\frac{d}{dy}\left(\frac{dy}{dx}\right)$$
when:
(i) $$y=x$$
(ii) $$y=x^2$$
(iii) $$y=x^3$$
(iv) $$y=x^n$$
(v) $$y=e^x$$
(vi) $$y=\sin(x)$$?

Differentiate this

$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$
Find $$f'(x)$$.

x to the power of x again

Let $$y=x^{x^{x^{x^{...}}}}$$ [$$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ($$x$$ to the power of ...))) with an infinite number of $$x$$s]. What is $$\frac{dy}{dx}$$?

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