# Tagged Questions

358 views

### Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$\sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx$$
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### Show that $\int_{\pi/4}^{\pi/2} \frac{\sin x}{x}\,dx\leq \frac{\sqrt{2}}{2}$

Show that $$\int_{\pi/4}^{\pi/2} \dfrac{\sin x}{x}\,dx\leq \dfrac{\sqrt{2}}{2}$$ Any Ideas, how to start ?!
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### A formula for $\pi$ and an inequality

For any $n\in \mathbb{N}$ prove the identity : $$\pi =\sum_{k=1}^{n}\frac{2^{k+1}}{k\dbinom{2k}{k}}+\frac{4^{n+1}}{\dbinom{2n}{n}}\int_{1}^{\infty}\frac{\mathrm{d}x}{(1+x^2)^{n+1}}\tag{1}$$ and thus ...
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### How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand?

Is there any way we can prove this definite integral inequality by hand: $$\int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4$$ I don't where to start even, please help. That ...
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### Differential equation with sec

With $(a)$ I got that $-y^2 dx = \sec^2x\ dy$, but it makes no sense. Hence, no Idea how to handle $(b)$.
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### If $f$ is $2 \pi$ periodic and $\int_{0}^{2 \pi} f(t) dt=0$ then $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$ [duplicate]

Given $f$ a real differentiable function, $2 \pi$ periodic such that $\int_{0}^{2 \pi} f(t) dt=0$ show that: $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$. When does equality hold? ...
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### Proof of integral inequality

How does one prove (without making use of any approximations whatsoever) the following inequality: $$\int_1^2 \left(\ln(x)\right)^{2013}dx\leq\dfrac{1}{2^{2013}}.$$
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### Compare the integrals $\int_0^{\frac{\pi}{2}}\sin(\cos x)dx$ and $\int_0^{\frac{\pi}{2}}\cos(\sin x)dx$

Compare the following two integrals: $$\int_0^{\frac{\pi}{2}}\sin(\cos x)dx,\quad \int_0^{\frac{\pi}{2}}\cos(\sin x)dx$$ First I observe that by making the change of variable $x=\frac{\pi}{2}-x$,we ...
How do I obtain upper and lower bounds for a summation function: $$\sum_{i=1}^{25}i^4$$ Somehow it involves an integral: $$\int_{0}^{25}x^4\:\mathrm{d}x$$ If you solve it it gives $1953125$ (this ...