Articles of calculus

A non-increasing particular sequence

For every fixed $t\ge 0$ I need to prove that the sequence $\big\{n\big(t^{\frac{1}{n}}-1\big) \big\}_{n\in \Bbb N}$ is non-increasing, i.e. $$n\big(t^{\frac{1}{n}}-1\big)\ge (n+1)\big(t^{\frac{1}{n+1}}-1\big)\;\ \forall n\in \Bbb N$$ I’m trying by induction over $n$, but got stuck in the proof for $n+1$: For n=2 its clear that follows since $$t-1\ge 2(t^{1/2}-1)\Leftrightarrow t-1\ge 2t^{1/2}-2\Leftrightarrow t+1\ge 2t^{1/2}\Leftrightarrow t^2+2t+1\ge 4t\Leftrightarrow t^2-2t+1\ge […]

$\lim_n \frac{b-a}{n}\sum_{k=1}^{n}\sup_{x\in}f$, $\lim_n \frac{b-a}{n}\sum_{k=1}^{n}\inf_{x\in}f$ and Darboux integrals

Let us use the notation $\overline{\int_a^b}f(x)dx$ for the Darboux upper integral of $f$ and $\underline{\int_a^b}f(x)dx$ for the lower one. Let us construct a partition of $[a,b]$ into $n$ intervals $[x_{k-1},x_k]$ defined by $x_k=a+k(b-a)/n$ and les us consider the corresponding Darboux sums$$\Delta_n=\frac{b-a}{n}\sum_{k=1}^{n}\sup_{x\in[x_{k-1},x_k]}f(x),\quad \delta_n=\frac{b-a}{n}\sum_{k=1}^{n}\inf_{x\in[x_{k-1},x_k]}f(x).$$ It is clear, by taking the definitions of $\sup$ and $\inf$, and the […]

calculate $\int_0^{\pi}\frac {x}{1+\cos^2x}dx$

Same to the tag calculate $\int_0^{\pi}\frac {x}{1+\cos^2x}dx$. Have no ideas on that. Any suggestion? Many thanks

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)\hspace{1mm}|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$ As you can see two forms are easy. $$\iiint_E \hspace{1mm}dV = \int_0^{1}\int_{x^2}^{\sqrt{x}}\int_0^{x+y} \hspace{1mm}dz\hspace{1mm}dy\hspace{1mm}dx = \int_0^{1}\int_{y^2}^{\sqrt{y}}\int_0^{x+y} \hspace{1mm}dz\hspace{1mm}dx\hspace{1mm}dy$$ After that I am having difficulty visualizing the 3D graph

How does Hessian matrix describe the local curvature?

For a 2-variable function $f(x,y)$, the Hessian matrix is $$\mathcal{H}(f) = \left[\begin{array}{cc} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial {xy}} \\ \frac{\partial^2 f}{\partial {xy}} & \frac{\partial^2 f}{\partial y^2}\end{array}\right]$$, how does it describe the local curvature of $f$?

Convergence, Integrals, and Limits question

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to prove that the series on the left size converges using the integral test, but now I’m having a hard time proving that the equality above is […]

Double integral $ \iint\ln(\sin(u-7v)) \,du\,dv$

Can you please help me solve this double integral? $$ \iint_D\ln(\sin(u-7v))\,du\,dv $$ where $$D:=\left\{(u,v)\;:\; 0\le u \le \pi \;,\; 0\le v \le \frac u7\right\} .$$ I know it’s not possible to solve $\int \ln(\sin(x))$ but definite integral is?

Proving $(1 + \frac{1}{n})^n < n$ for natural numbers with $n \geq 3$.

Prove with induction on $n$ that \begin{align*} \Bigl(1+ \frac{1}{n}\Bigr)^n < n \end{align*} for natural numbers $n \geq 3$. Attempt at proof: Basic step. This can be verified easily. Induction step. Suppose the assertion holds for $n >3$, then we now prove it for $n+1$. We want to prove that \begin{align*} \big( 1+ \frac{1}{n+1})^{n+1} < n+1. […]

Integral help with completing the square!!

Could you please help me integrate $$\int\frac{dx}{\sqrt{2+x-x^2}}$$ I am supposed to complete the square.. but I am seriously stuck. I have tried to square each side and stuff, but it does not seem to be working

Some pecular fractional integrals/derivatives of the natural logarithm

(Be prepared for a very long post) I have deduced the following formula: $$D^{-n}\ln(x)=\frac{x^n(\ln(x)-n)}{(-n)!}=\frac{x^n(\ln(x)-n)}{\Gamma(-n+1)}$$ Where $$D^{-1}f(x)=\int f(x)dx$$ $$D^{-2}f(x)=\int\int f(x)dxdx$$ $etc$. $$D^0f(x)=f(x)$$ $$D^1f(x)=\frac d{dx}f(x)$$ $$D^nf(x)=\frac{d^n}{dx^n}f(x)$$ So the $n$th integral of $\ln(x)$ is given by my formula if $n$ is a natural number. Since the formula is continuous for $n\in\mathbb{R}$ when $n$ is not a negative integer, […]