5
votes
4answers
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): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
0
votes
1answer
73 views

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 ?!
6
votes
1answer
76 views

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 ...
6
votes
4answers
152 views

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 ...
1
vote
2answers
51 views

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 $(b)$.
5
votes
0answers
38 views

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? ...
2
votes
1answer
88 views

Proof of integral inequality

How does one prove (without making use of any approximations whatsoever) the following inequality: $$\int_1^2 \left(\ln(x)\right)^{2013}dx\leq\dfrac{1}{2^{2013}}.$$
4
votes
1answer
114 views

Integral inequality with a function twice differentiable

Let $f:[0,1]\longrightarrow\mathbb{R}$ be a function twice differentiable with continous second derivative and $f(1)=f(0)$. The inequality: $$\int_{0}^{1}(f''(x))^2dx\geq ...
1
vote
1answer
22 views

Help with integral/logarithm inequality

I have to prove the following inequality: $1/(n+1) < \int_n^{n+1} 1/t$ $dt$ $<1/n$ I thought it would be easier to attack this via integration, so I get: $1/(n+1) <$ log $(n+1)-$ ...
0
votes
1answer
56 views

Validity of an integral inequality

Suppose we have two functions $f(x)$ and $g(x)$ And $f(x)<g(x)$ for all values of $n$ Then for arbitrary $a$ and $b$ (within the range) is it true that $$\int_b^{a}\frac{dx}{f(x)} > ...
7
votes
1answer
130 views

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 ...
2
votes
1answer
190 views

Obtain lower and upper bounds

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 ...
1
vote
1answer
68 views

Difference of definite integrals inequality

Could you help me how prove that for any $\mathcal{C}^1$ function we have: $$\left|\int_{a} ^{\frac{a+b}{2}}f(x) d x - \int_{\frac{a+b}{2}} ^bf(x)dx\right| \le \frac{(b-a)^2}{4} \cdot \max _{x \in ...
10
votes
1answer
206 views

Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that $$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$ Prove that $$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$ Where should ...
17
votes
1answer
191 views

How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?

I want to prove the inequality $$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$ There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
2
votes
4answers
146 views

prove or disprove that $\int_a^b |f(x)| \mathrm{d}x\geq |\int_a^b f(x)\mathrm{d}x |$

+Let $f$ be a continuous and integrable function over $[a;b]$, Prove or disprove that : $\displaystyle\int_a^b |f(x)|\ \mathrm{d}x\geq \left | \int_a^b f(x)\ \mathrm{d}x\right| $
1
vote
1answer
143 views

Show that $\int_0^1f(x)^3dx\le\left(\int_0^1f(x)dx\right)^2$ [duplicate]

Possible Duplicate: Prove that $\int_0^x f^3 \le \left(\int_0^x f\right)^2$ Let $f$ be a differentiable function on $[0,1]$. $f(0)=0$ and $1\ge f'(x)\ge0$. Show that ...
9
votes
1answer
357 views

Prove that $\int_0^1|f''(x)|dx\ge4.$

Let $f$ be a $C^2$ function on $[0,1]$. $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that $\int_0^1|f''(x)|dx\ge4.$ Also determine all possible $f$ when equality occurs.
8
votes
1answer
312 views

Integral-Summation inequality.

The following question was in an entrance exam: Show that, if $n\gt0$, then: $$\int_{{\rm e}^{1/n}}^{\infty}{\frac{\ln{x}}{x^{n+1}}\:dx}=\frac{2}{n^2{\rm e}}$$ You are allowed to assume ...
10
votes
3answers
374 views

Prove that $ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$

Prove the following integral inequality: $$ 1.462 \le \int_0^1 e^{{x}^{2}}\le 1.463$$ This is a high school problem. So far i did manage to prove that the integral is bigger than $1.462$ by using ...
8
votes
1answer
574 views

Prove that: $ \int_{0}^{1} \ln \sqrt{\frac{1+\cos x}{1-\sin x}}\le \ln 2$

I plan to prove the following integral inequality: $$ \int_{0}^{1} \ln \sqrt{\frac{1+\cos x}{1-\sin x}}\le \ln 2$$ Since we have to deal with a convex function on this interval i thought of ...
3
votes
3answers
145 views

Squeeze an integral

Would you have any idea about this problem ? Prove that for all nonnegative integers $n$, the following inequalities hold: $$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...