# Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.

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### Prove $\int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n}$

Nice little generalization: $$\int_0^1 \frac{\ln^a(1-x)\ln(1+x)}{x}dx=(-1)^a a! \sum_{n=1}^\infty\frac{H_n^{(a+1)}}{n2^n},\quad a=0,1,2,...$$ The point of this post is to save us some calculations ...
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### Symbolic approximation through integration by parts

This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt$ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get ...
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### Juantheron-like integral

While seeing this post, the following integral is just struck me $$\int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1$$ I have tried like what user @OlivierOloa did in his ...
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### Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
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### Trying to prove a proposition about the nth order derivative of a polynomial by induction - is this correct?

Recently, I decided to try and create a formula for the $n$th order derivative of a polynomial, and I believe I succeeded! I tried to do a proof by induction to confirm this for myself, but since I ...
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### Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Given the Elliptic integral singular value $K(k_m)$, Dedekind eta $\eta(\tau)$, j-function $j(\tau)$, and hypergeometric $_2F_1\left(a,b;c;z\right)$ with $\color{brown}{a+b=c=\tfrac12}$. Conjecture: &...
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### Can we further refine $\int_{0}^{1}x^{x^x} \ dx=\frac 1 2+\sum_{n=1}^{\infty}(-1)^n\sum_{k=1}^{\infty}\frac {(n-k)^k}{(k+1)^{n+1}} \binom {n}{k}$

Question Can we further refine the integral $$\int_{0}^{1}x^{x^x}\ dx=\frac 1 2+\sum_{n=1}^{\infty}(-1)^n\sum_{k=1}^{n}\frac {(n-k)^k}{(k+1)^{n+1}} \binom {n}{k}$$ ? To compute the result, first ...
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### Is it feasible to use Operator Calculus to solve for $f(t)=\underbrace{\exp(\exp(\dots\exp}_{t}(0)\dots))$ over $\mathbb{R}$

Consider the function $f(t)=\exp^{(t)}(0)$ where $\exp^{(0)}(0)=0$ and $\exp^{(t+1)}(0)=\exp(\exp^{(t)}(0))$. That is, $$f(t)=\underbrace{\exp(\exp(\dots\exp}_{t}(0)\dots)).$$ Such a function is not ...
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### Why are these integrals equivalent to the Riemann Hypothesis?

Riemann Hypothesis is equivalent to the integral equation $$\int_{-\infty}^{\infty} \frac{\log \mid \zeta (1/2+it)\mid }{1+4t^2} \ dt =0$$ Many other integral equations exist that are equivalent. How ...
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### Strong Counterexample to MVT on Q

A well-known application of the MVT is to prove that any $f: \mathbb{R} \to \mathbb{R}$ with $f'= 0$ is constant. But of course, the MVT relies fundamentally on the properties of $\mathbb{R}$, and in ...
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### Evaluate $\sum_{n=0}^{\infty} \frac{n!}{(n^2)!}$

I'm interested in a method of evaluating $\sum_{n=0}^{\infty} \frac{n!}{(n^2)!}$. If there was a linear equation with leading coefficient $1$ in the denominator or a quadratic with leading ...
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### Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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