Found problems: 50
Let be a natural number $ n, $ a nonzero number $ \alpha, \quad n $ numbers $ a_1,a_2,\ldots ,a_n $ and $ n+1 $ functions $ f_0,f_1,f_2,\ldots ,f_n $ such that $ f_0=\alpha $ and the rest are defined recursively as
$$ f_k (x)=a_k+\int_0^x f_{k-1} (x)dx . $$
Prove that if all these functions are everywhere nonnegative, then the sum of all these functions is everywhere nonnegative.
Calculate the integral $$\int \frac{dx}{\sin (x - 1) \sin (x - 2)} .$$
Hint: Change $\tan x = t$ .
Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]
Evaluate the following limit.
\[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]
Find the value of the integral $\int_{-1}^{1} ln \left(x +\sqrt{1 + x^2}\right) dx$.
Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that:
$$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$
Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy
$$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$
for any nonnegative integer $ n. $
[i]Mihai Piticari[/i]
Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$,
we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.
Let be two real numbers $ a<b $ and a function $ f:[a,b]\longrightarrow\mathbb{R} $ having the property that if the sequence $ \left(f\left( x_n \right)\right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent.
[b]a)[/b] Prove that if $ f $ admits antiderivatives, then $ f $ is integrable.
[b]b)[/b] Is the converse of [b]a)[/b] true?
[i]Marcelina Popa[/i]
Let $f:\mathbb{R}\rightarrow (0,\infty)$ be continuous function of period $1$. Prove that for any $a\in\mathbb{R}$ $$\int_0^1\frac{f(x)}{f(x+a)}dx\geq 1.$$
Calculate the definite integral
$$\int_2^4 \sin ((x-3)^3) dx$$
Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that
$$ \left| \int_0^2 f(t) dt\right| >1. $$
Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$
[list=a]
[*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$
[*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$
[/list][i]Mihai Piticari and Sorin Rădulescu[/i]
Let $ f:[-1,1]\longrightarrow\mathbb{R} $ a derivable function and a non-negative integer $ n. $ Show that there is a $ c\in [-1,1] $ so that:
$$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$
Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$
$\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$
$\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$
[i]Nicolae Bourbacut[/i]
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x = 0$ of $(1 + x)^{\alpha}$ . Evaluate
$$\int_{0}^{1} \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\, dy.$$
Let $ f,g:[0,1]\longrightarrow{R} $ be two continuous functions such that $ f(x)g(x)\ge 4x^2, $ for all $ x\in [0,1] . $ Prove that
$$ \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1. $$
Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying
$$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$
for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e.
$$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$
[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $
[b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $
[i]Florin Stănescu[/i]
Prove that if $f(x)$ is continuous for $a\leq x \leq b$ and
$$\int_{a}^{b} x^n f(x) \, dx =0$$
for $n=0,1,2, \ldots,$ then $f(x)$ is identically zero on $a \leq x \leq b.$
For any partition $ P $ of $ [0,1] $ , consider the set
$$ \mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f’\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} . $$
Prove that there exists a partition $ P_0 $ of $ [0,1] $ such that
$$ g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g’(x)\big| >4\cdot \# P. $$
Here, $ \# D $ denotes the natural number $ d $ such that $ 0=x_0<x_1<\cdots <x_d=1 $ is a partition $ D $ of $
[0,1] . $
Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.
Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous and nondecreasing function.
[b]a)[/b] Show that the sequence $ \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} $ is nonincreasing.
[b]b)[/b] Prove that, if there exists some natural index at which the sequence above is equal to $ \int_0^1 f(x)dx, $ then $ f $ is constant.
Let $f:[0,1]\to(0,\infty)$ be a continous function on $[0,1]$ and let $A=\int_0^1 f(t)\mathrm{d}t.$[list=a]
[*]Consider the function $F:[0,1]\to[0,A]$ defined by \[F(x)=\int_0^xf(t)\mathrm{d}t.\]Prove that $F(x)$ has an inverse function, which is differentiable.
[*]Prove that there exists a unique function $g:[0,1]\to[0,1]$ for which\[\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}t\]is satisfied for every $x\in [0,1].$
[*]Prove that there exists $c\in[0,1]$ for which\[\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,\]whre $g$ is the function uniquely determined at b.
[/list]
Evaluate the integrals as follows.
(1) $\int \frac{x^2}{2-x}\ dx$
(2) $\int \sqrt[3]{x^5+x^3}\ dx$
(3) $\int_0^1 (1-x)\cos \pi x\ dx$