# Tagged Questions

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### What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$\frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0$$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
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### $\int xtanx$ and the Clausen Function

I have been attempting to evaluate $\int x \tan x \;\mathrm{d} x$. My first instinct was integration by parts, which produces $-x \ln|\cos x|+\int \ln|\cos x| \;\mathrm{d} x$. I have read online ...
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### Interpolate the number of arrangements in a set

I am working the integral $$\int_0^\infty e^{x(k-\alpha) - e^x} dx$$ where $k$ is a positive integer and $\alpha$ a positive real. WolframAlpha shows that for $\alpha=0$ and $k=1,\ldots,7$ the ...
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### Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
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### Fourier transform of $\operatorname{erfc}^2\left|x\right|$

Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^2\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the complementary error function.
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### How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and ...
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### How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ dx$?

I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ but have problems ...
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### Why has $\int \sin (\sin x) dx$ not been solved yet?

I have Calculus 2 background, so please try to keep your answers around that level. I inly want a brief explanation. What is it about $\sin (\sin x)$ that makes it difficult to integrate? Also, what ...
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### Inverse of the Modified Bessel function

Is there any chance of having a formula or approximation to inverse the Modified Bessel function of the first kind? I mean to solve $I_M(x)$ with respect to x: $I^{-1}_M(x)$? Thanks in advance
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### help with complicated modified bessel function integral

I am trying to address the following complicated integral $$\int_0^{\infty} x^{m-1} e^{-(ax^2+bx+c)}I_v(kx)\text{d}x,$$ Where $I_v(x)$ is a modified Bessel function of the first kind. I did try to ...
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### How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the ...
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### Integral with incomplete gamma function and Modified Bessel Function

Can somebody suggest a technique to integrate this? all parameters (m, beta, v, k, A, B and n) are positive real constants
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### Closed form for derivative $\frac{d}{d\beta}\,{_2F_1}\left(\frac13,\,\beta;\,\frac43;\,\frac89\right)\Big|_{\beta=\frac56}$

As far as I know, there is no general way to evaluate derivatives of hypergeometric functions with respect to their parameters in a closed form, but for some particular cases it may be possible. I am ...
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### Interesting integral with Modified Bessel Function, Gamma Function

Is there anyway to integrate this monster? m, beta, v, k, A, B, and n are real positive constants.
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### Gaussian-Like integral

What is the integral of this? $$\int_0^\infty xe^{-(ax^2+bx)}\,\mathrm{d}x$$ $a$ and $b$ are positive integers.
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### How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
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### Taylor series of some stange function

Can somebody help find the Taylor series for: where p and n are real positive (not necessary integers)? Does it converge fast?
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### Does $\int^1_0\frac{(1-s)^\alpha}{s^\beta} \operatorname d \!s$ converge?

I want to ask about the finitess of the following integral $$\int^1_0\frac{(1-s)^\alpha}{s^\beta}ds$$ when $\alpha>\beta>1$. This integral is very similar to the ...
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### Closed form integral $\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$

Is there a closed form expression for the definite integral $$I=\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$$ for $a<b<c<d$? Mathematica 9.0 can do it for special cases using ...
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### integral with gaussian function

I am trying to evaluate the following integral: $$\int_0^\infty{z^{m-1}\over\left[1+\left(\eta z\right)^n\right]^p}e^{-(z-b)^2\over c}\,{\rm d}z,$$ where the integration is w.r.t. to $z$, and the ...
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### integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
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### Is there any infinite series representation of the sine integral?

Is there any infinite series representation of the sine integral? It is defined as $$\displaystyle \int\ \frac{\sin(x)}{x}\ dx$$
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### Could a computer theoretically compute all integrals in terms of some special functions or it is not possible theoretically?

Could a computer theoretically compute all integrals in terms of some special functions or there need to be exist infinite number of such special functions to represent all integrals? I know there ...
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### Fourier transform of Legendre

I am trying to figure out the Fourier transform of Legendre polynomial $P_\ell [\cos(\theta-a t )]$: $Q(\omega)=\int_{-\infty}^\infty P_\ell [\sin\phi\cos(\theta-a t )] e^{i \omega t} dt,$ where ...
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### $\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$ [duplicate]

How to Find $$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$$ the most basic way possible? I know it's $\pi^4/90$ but to arrive at this figure? Curious, because I need it to solve the integral ...
Can the heaviside step function, $u(t)$ be represented like so: $$u(t)=\frac{1}{2}\left(\frac{|x|}{x}+1\right)$$
Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...