Tagged Questions
1
vote
1answer
46 views
The integral $\int_0^1\dfrac{(-x)^n}{1+x} dx $
How can I prove that:
$\forall x \in \mathbb{N}\setminus {0} \quad \dfrac{-1}{n+1}\le \int_0^1\dfrac{(-x)^n}{1+x} dx \le \dfrac{1}{1+n}$
$\lim_{n\to+\infty}\Sigma_{i=1}^{n}\dfrac{(-1)^{i-1}}{i}$.
...
2
votes
2answers
160 views
Prove $|\int_a^b$$f(x)dx| \leq \int_a^b$$|f(x)|dx$
Prove $$\left|\int_a^b f(x)dx\right| \leq \int_a^b |f(x)|dx.$$
My thoughts: first I think we must show that if $f \geq 0$ is Riemann integrable on $[a,b]$, then $\int_a^b f(x)dx \geq 0$. Then we ...
2
votes
2answers
56 views
Need to prove $\frac{3}{5}(2^{\frac{1}{3}}-1)\le\int_0^1\frac{x^4}{(1+x^6)^{\frac{2}{3}}}dx\le1$
I need to show that
$$\frac{3}{5}(2^{\frac{1}{3}}-1)\le\int_0^1\frac{x^4}{(1+x^6)^{\frac{2}{3}}}dx\le1$$
I just know that if in $[a,b]$, $f(x)\le g(x)\le h(x)$, then
...
2
votes
2answers
104 views
integration inequality [duplicate]
Possible Duplicate:
Proving Integral Inequality
Suppose $f(x)$ is differentiable on $[0,1]$ , $f(0)=0$ and $1\geq f'(x) >0 $
Prove that $\displaystyle\left(\int_{0}^{1} ...
9
votes
1answer
167 views
Prove the following integral inequality
Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality:
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17
votes
1answer
588 views
Do inequalities that hold for infinite sums hold for integrals too?
Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on ...
3
votes
1answer
139 views
Prove that $\int_0^1f(x)dx$$\int_0^1g(x)dx\ge1$.
Suppose $f(x)$ and $g(x)$ are positive measurable functions defined on $(0,1)$, satisfying $f(x)g(x)\ge1$ for any $x\in(0,1)$. Prove that $\int_0^1f(x)dx$$\int_0^1g(x)dx\ge1$.
Totally no idea about ...
