# Tagged Questions

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### Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$

Prove that: $(1)$$\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \ dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \ dx$$$(2)$$\int_0^{\infty } \frac{1}{\sqrt{8 x^3+x+7}} \ dx>1$$ What I do for ...
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In Abramowitz and Stegun: Handbook of Mathematical Functions (on page 229, property 5.1.20) it is found that $$\frac{1}{2} \log \left(1 + \frac{2}{x} \right) < \exp(x) E_1(x) < \log \left(1 + ... 1answer 84 views ### Show g(\mathbf{x}) \leq h(\mathbf{x}) implies \int g(\mathbf{x})\mathrm{d}\mathbf{x} \leq \int h(\mathbf{x})\mathrm{d}\mathbf{x} Suppose I have g and h from \mathbb{R}^p\to\mathbb{R} such that for all \mathbf{x}, g(\mathbf{x}) \leq h(\mathbf{x}). I want to prove that the integral over all \mathbb{R}^p of g is less ... 1answer 18 views ### Simplification of Equation Involving Second Partials I was reading this article and I'm trying to follow this author's proof. The author jumps from$$\psi_1(x)\frac{\partial^2\psi_2(x)}{\partial x^2}-\psi_2(x)\frac{\partial^2\psi_1(x)}{\partial ...
Prove that $\sum_{n=2}^{\infty} \frac{z^{n-1}}{\alpha(n-1)+1}$ is equivalent to $\frac{1}{\alpha} \displaystyle \int_{0}^{1}{ \frac{z t^{\frac{1}{\alpha}}}{1-tz}} dt$?