# Questions tagged [integration]

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.

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### Computing the limit $\lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x$ for fixed $n \in \mathbb{N}$

I'm working on a problem that asks to compute $$\lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x$$ for fixed $n \in \mathbb{N}$. What I've tried so far is to do a $u$-...
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### Integration of $3x^2ydx+x^3dy$ by two methods

Find $\int(3x^2ydx+x^3dy)$ $\int(3x^2ydx+x^3dy)=\int d(x^3y)=x^3y+c$ Or, $\int(3x^2ydx+x^3dy)=\int3x^2ydx+\int x^3dy=x^3y+x^3y+c=2x^3y+c$ Why am I getting different answers from the two methods? I am ...
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### Positive definite Hessian matrix

1)If I want to prove a Hessian matrix is positive definite and it contains the entries of $x=(x_1,...,x_n)$, can I multiply it also with the same $x$ to simplify my calculations? 2)Is the integral of ...
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### Verify the inequality without evaluating the integrals [closed]

Use the properties of integrals to verify the inequality without evaluating the integrals. Explain why. (Use the graph. $f(x)=x^2+1$) $$\cfrac{3}{4}\le\int_0^1{e^{x^2}}\,\mathrm dx\le{e}$$ I just ...
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### Infinite sum of iterated integrals of matrix products

The problem: Let $$N(z) = \begin{pmatrix} 0 & \frac{1}{z} \\ \frac{1}{2(z-1)^2} & 0 \end{pmatrix} \;.$$ I write this post to seek any hints about how to compute the following infinite sum over ...
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### Different notation in Expectation's integration in (Empirical) Risk Minimization

For the philosophy of Statistical Learning Theory, the Risk Minimization is the focus. We want to minimize $$R(f)=\mathbb{E}[L(f(x),y)]=\int~L(f(x),y) ~dP(x,y),$$ where $P(x,y)$ is the "joint ...
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### Given a general region evaluate the double integral over that region.

Consider the integral $$J = \iint_T \log(1+x^2)\,dx \,dy,$$ where $T$ is the region that is bounded $y=0$, $x=y$ and $x=2$. My attempt: $$\int^{x=2}_{x=0} \, \int^{y=x}_{y=0} \log(1+x^2) \,dy\,dx$$ ...
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### Prove if we have for all $0 \leq k \leq n, \, \int_0^1 x^k f(x) \, \mathrm{d}x =0$ then $f$ has at least $n+1$ zeroes.

Let $n\in \mathbb{N}$ and $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that, $$\forall 0 \leq k \leq n, \,\int_0^1 x^kf(x) \, \mathrm{d}x=0$$ prove that $f$ has at least $n+1$ zeroes. What I've done ...