# All Questions

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### Computing $\int \dfrac{x^4}{x^3-8} dx$

I am currently stuck with one integration and I don't know what to do know. I would appreciate detailed answer, but will be happy for any :-) I am having: $$\int \dfrac{x^4}{x^3-8} \,dx$$ I am ...
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### Continuous functions vanishing at infinity is always integrable?

Let $$C_{0}(\mathbb R)= \big\{\,f:\mathbb R \to \mathbb C\,\,\, \text{continuous and}\,\, \lim_{x\to \pm \infty}f(x)=0 \big\}.$$ Assume that $f\in C_{0}(\mathbb R)$. My question is: Is it always ...
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### Geometric interpretation of Euler's identity for homogeneous functions

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is called homogeneous of degree $d \geq 0$ if $$f(\lambda x_1, \ldots, \lambda x_n ) = \lambda^d f(x_1, \ldots, x_n)$$ Differentiating both sides ...
### Nonsingular projective variety of degree $d$
For each $d>0$ and $p=0$ or $p$ prime find a nonsingular curve in $\mathbb{P}^{2}$ of degree $d$. I'm very close just stuck on one small case. If $p\nmid d$ then $x^{d}+y^{d}+z^{d}$ works. If ...