For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.
Questions tagged [integral-inequality]
1106 questions
0
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Inequality and integrals
I want to prove that:
$$\int_0^1\frac{2\sin{x}}{x^2+1}\text{d}x\le\ln{2}$$
So I did the followings:
$$\int_0^1\frac{2\sin{x}}{x^2+1}\text{ d}x\le\ln{2}\iff\int_0^1\frac{2\sin{x}}{x^2+1}\text{ d}x\le\int_0^1\ln{2}\text{…
Leos Kotrop
- 1,195
-1
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1 answer
How prove this integral inequality $\int_{0}^{1}x \sqrt{1+\{f'(x)\}^2}dx\le\ \frac{1}{\sqrt{2}}$
Let $f$ is two differentiable function on $[0,1]$ such that $f(0)=1$, $f(1)=0$, $f'(x)\le 0$, $f''(x)\ge0$
Show that
$$\int_{0}^{1}x \sqrt{1+\{f'(x)\}^2}dx\le\ \frac{1}{\sqrt{2}}$$
How do have it? Thank you.
JW Dark
- 3
-1
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1 answer
Prove inequality of natural x, y, z
Prove that, for every x , y, z of the natural inequality is true:
$x\ $ $+$ $y\ $ $+$ $z\ $ $\ge$ $3$ $and$ $x$*$y$*$z$ $=$ $1$
I tried to divide by 3, but does not go :(
JWa
- 59
-2
votes
1 answer
Inequality involving the definite integrals of cos(x) f(x) and sin(x) f(x)
Considering this trivial definite integrals inequality:
$$
\mbox{with} \;a \in (0, 2\pi)\; \mbox{such that} \;\; \int_{0} ^{a} \cos(x) dx \; = \; 0 \; \; \; \;\Longrightarrow \; \; \; \; \int_{0} ^{a} \sin(x) dx \; \neq \; 0
$$
Inserting now a…
Luca
- 182