# Tagged Questions

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### Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?

In evaluating integrals like (link to another example) $$I=\int_0^1\frac{\log(x) \log^2(1-x)dx}{x}$$ one can make the substitution $x=\sin^2(\theta)$ to obtain ...
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### Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have ...
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### An identity involving integrals of the form $\int_{0}^{\alpha} \arctan (\sqrt{5}\tan\theta) \, d\theta$.

I want to establish the following identity $$\int_0^{\pi/3} \arctan (\sqrt{5} \tan \theta) \, d\theta - 2 \int_0^{\pi/6} \arctan (\sqrt{5} \tan \theta) \, d\theta = \frac{\pi^{2}}{30}. \tag{1}$$ ...
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### Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
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### Is there a closed form for this sum?

While generalizing the previous result, I conjectured that the series expansion of \begin{align*} \int_{0}^{\frac{\pi}{2}} \arctan \left( \frac{2x \sin\theta}{1-x^{2}} \right) \arctan \left( \frac{2y ...
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### An integral relating to Bernoulli polynomials

Show that $$\int_{0}^{1}B_{2n+1}(x)(\cot({\pi}x)-2\sin(2{\pi}x))dx{\sim}0$$ where $B_{2n+1}(x)$ is the Bernoulli polynomials.
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### Bernoulli number type asymptotics

I find an interesting formula but I can not prove it. Show that $$I_n=(-1)^{n+1}\int_0^1 B_{2n+1}(x)\cot(\pi x) \, dx\sim\frac{2(2n+1)!}{(2\pi)^{2n+1}}$$ where $B_n(x)$ is the Bernoulli Polynomials.
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### Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx = -\frac{\pi^3}{48}-\frac{\pi}{8}\log^2 2 +G\log 2$$ where $G$ is the Catalan's Constant. Numerically, it's ...
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### Integral of the Planck radiation formula

It's well known that the blackbody radiation law has ben derived by Plank and its mathematical formula is: $$W(\lambda,T)=\frac{C_1}{\lambda^5\left(\exp\frac{C_2}{\lambda T}-1\right)}$$ The definite ...
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### Methods for computing finite part integrals when corresponding convergent integrals are known

Frequently when integrating special functions, only certain definite integrals are known and the elementary antiderivatives cannot be found. This makes computing Cauchy Principal Value and other ...
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### Integration gamma and beta: $\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$

How can we evaluate the following integral? $$\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$$ I can't find anything to substitute because all of the trigonometric identities are in square form...
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### Does this integral have a closed form: $\int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y}$?

Consider the following integral: $$G(\beta,\delta,y) = \int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y},$$ with $\delta>0$, $\Re\beta>0$, $y\neq1$. Does it have a closed form in ...
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### Integral identity with square of Jacobi polynomial

This has stumped me for a while: I have a function $\zeta_k^S(x)$ that can be expressed using Jacobi polynomials $P_k^{(\alpha,\beta)}(x)$: ...
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### Show that $\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$

The question asks to prove the identity: $$\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$$ where $n\in\mathbb{Z}$ I have no idea ...
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### Integral involving exponential, power and Bessel function

Is there any formula for calculating the following definite integral, including exponential and Bessel function? $$\int_0^{a}x^{-1} e^{x}I_2(bx)dx$$ Thanks in advance
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### Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
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### Integral with Bessel functions of the First Kind.

I'd like to solve the following integral: $I = \int_0^\infty J_0(at) J_1(bt) e^{-t} dt\$ where $J_n$ is an $n^{th}$ order Bessel Function of the First Kind and $a$ and $b$ are both positive real ...
### Rewriting the integral $\mathrm{erf}(x) = \frac{1}{\sqrt{\pi}} \int_{-x}^x e^{-t^2} dt.$
The Legendre Polynomials satisfy the following orthogonality condition: The definite integral of $P(n,x) \cdot P(m,x)$ from $-1$ to $1$ equals $0$, if $m$ is not equal to $n$: \int_{-1}^1 P(n,x) ...