# Tagged Questions

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### Find all differentiable functions $f:[0;2] \to \Bbb{R}$ such that $\int_{0}^{2}xf(x)dx=f(0)+f(2)$

Find all differentiable functions $f:[0;2] \to \Bbb{R}$, with $f'$ continuous, such that the function $e^{-x}f(x)$ is decreasing on $[0;1]$ and increasing on $[1;2]$, and ...
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### Let $f: \Bbb{R} \to [0; \infty)$ .Prove that $\forall n \in \Bbb{N}$ $\forall y \in \Bbb{R}$ $\exists t=t(n;y)$ such that $\int_{y}^{t}f(x)dx=n$

The problem goes like this: Let $f: \Bbb{R} \to [0; \infty)$ be a continuous function such that $\lim_{x \to \infty}f(x)=\infty$. Prove that $\forall n \in \Bbb{N^{*}}$ and $\forall y \in \Bbb{R}$ ...
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### The Monster PolyLog Integral $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$\int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x$$ which is from some high school training ...
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### Integral $\int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$

I am trying to solve this integral $$\int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx$$ A closed form does exist despite the looks of the integrand. ...
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### Integral $I=\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx$

Hi I am stuck on showing that $$\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx=\pi G-\frac{3\zeta(3)}{8}$$ where G is the Catalan constant and $\zeta(3)$ is the Riemann zeta function. Explictly ...
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### Integral $6\int_{x=0}^{x=1}\int_{y=x}^{y=1}\int_{z=x}^{z=y} f(x) f(y) f(z)dxdydz=\bigg(\int_0^1 f(t) dt\bigg)^3$

Prove that $$6\int_{x=0}^{x=1}\int_{y=x}^{y=1}\int_{z=x}^{z=y} f(x) f(y) f(z)dxdydz=\bigg(\int_0^1 f(t) dt\bigg)^3$$ assuming $f(x)$ is continuous on [0,1]. This is from an old Putnam exam. I am ...
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### Integral $I=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0.$

$$I(\alpha,\beta)=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0.$$ I am trying to solve this integral. This is from the old high school ...
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### Putnam Exam Integral

I am trying to evaluate$$\lim_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+...x_n)\big)dx_1 dx_2...dx_n.$$ This is from an old Putnam mathematics competition. Either ...
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### Integrating $\int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx.$

Compute $$\int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx.$$ I am not sure how to start this one...I am thinking of a substitution to get started.
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### Integral $\int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$

Hello there I am trying to calculate $$\int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$$ NOT using mathematica, matlab, etc. We are given that $\sigma, \omega$ are complex. Note, the ...
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### Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}.$$ Note, we can expand the log in the integral to ...
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### Integral, Definite Integral $\int_{-\infty}^\infty \exp{\big(\alpha x^4+\beta x^3+\gamma x^2 +\delta x+\epsilon}\big)dx, \ \alpha <0.$

Calculate the integral $$I=\int_{-\infty}^\infty \exp{\big(\alpha x^4+\beta x^3+\gamma x^2 +\delta x+\epsilon}\big)dx, \ \alpha <0.$$ The answer can be expressed analytically in terms of a ...
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### Integral $\int_0^{\pi/2} x\cot(x)dx$, Differntiation wrt parameter only.

Integrate using differentiation wrt parameter only. $$\int_0^{\pi/2} x\cot(x)dx$$ We can express this as $$\int_0^{\pi/2} x\cdot\frac{\cos(x)}{\sin(x)}dx$$ Notice we can write $u=\sin(x)$ ...
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### different wrt parameter $I=\int_0^\infty \frac{1}{(x^2+p)^{n+1}}dx$

Integrate using differentiation with respect to parameter only: $$I=\int_0^\infty \frac{1}{(x^2+p)^{n+1}}dx, \ n\geq 0, \ p\geq1$$ No complex methods allowed. This is a rather useful integral to ...
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### Computing the integral $\int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi.$

Integrate $$\int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2) \, d\phi.$$ Something that may help $(1-2x\cos\phi+x^2)=(1-xe^{i\phi})(1-xe^{-i\phi})$. And using the series ...
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### Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$\int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx$$ No complex variables, only this approach. Interesting ...
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### definiteinteggral

The integral is given by $$\int_0^1 \frac{\ln (1-x)\ln x}{1+x} dx = \frac{1}{8}\big(-\pi^2\ln(4) +13\zeta(3)\big).$$ Any ideas how to prove? We cannot solve the integral so easily because we cannot ...
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### Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
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### Prove that $0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$

Let $f'$ be integrable. Prove that $$0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$$ Source: ...
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### Compute $\int_0^1\int_0^1…\int_0^1\lfloor{x_1+x_2+…+x_n}\rfloor dx_1dx_2…dx_n$

Compute $\int_0^1\int_0^1...\int_0^1\lfloor{x_1+x_2+...+x_n}\rfloor dx_1dx_2...dx_n$ where the integrand consists of the floor (or greatest integer less than or equal) function. The case $n=1,2,3$ ...
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### Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.

Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$ A start: If ...
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### Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}dx$

A friend of mine sent me an integral that she had not been able to crack, and me neither. It comes with a result, but without a proof (I suppose it originated in some math contest). Could you please ...
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### How to find $\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$?

Integration by parts is of no success. What else to try? $$\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$$
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### How to verify method used to solve integral was actually the fastest?

Is there any way to verify if the method I chose to integrate (by hand) was indeed fastest, or if there exists some better technique? Can a computer tell me or show me what the fastest method was, ...
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### An integral involving the error function

I have in my notes the following problem. I recall it being quite difficult and needing a change of variables into polar or spherical coordinates. Assuming I have not made a typo, there is a nice ...
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### Trying to recall an integration trick

In my notes, I have the following problem. Find the volume of (a) $x^2+y^2 \le 1$, $x^2+z^2\le 1$ in $\mathbb R^3$ (b) $x^2+y^2 \le 1$, $x^2+z^2\le 1$, $y^2+z^2\le 1$ in $\mathbb R^3$ ...
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$$\int\cos x\cdot\cos^2(2x)\cdot\cos^3(3x)\cdot\cos^4(4x)\cdot\ldots\cdot\cos^{2002}(2002x)dx$$ Taken from the 2002 Romanian olympiad
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### Integrals from MIT integration bee

$\int\frac{dx}{2+2\sin x+\cos x}$ $\int_0^{\infty}\frac{\ln x}{1+x^2}dx$ $\int\frac{dx}{x(1+x^3)}$ In general what is $\int \frac{dx}{a+b\sin x}$?
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### how prove this integral inequality?

How prove that for all continuous and decreasing function $f:[0 ,1]\mapsto(0,+\infty)$ $$\frac{\int_{0}^1x(f(x))^2dx}{\int_{0}^1xf(x)dx}\leq \frac{\int_{0}^1(f(x))^2dx}{\int_{0}^1f(x)dx}$$ thanks in ...