Found problems: 913
Calculate the following indefinite integrals.
[1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$
[2] $\int \frac{e^x}{e^x+e^{a-x}}dx$
[3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$
[4] $\int x\ln (x^2-1)dx$
[5] $\int \frac{2(x+2)}{x^2+4x+1}dx$
Calculate the following indefinite integrals.
[1] $\int \frac{(x^2-1)^2}{x^4}dx$
[2] $\int \frac{e^{3x}}{\sqrt{e^x+1}}dx$
[3] $\int \sin 2x\cos 3xdx$
[4] $\int x\ln (x+1)dx$
[5] $\int \frac{x}{(x+3)^2}dx$
Given that $ a,b,c, x_1, x_2, ... , x_5 $ are real positives such that $ a+b+c=1 $ and $ x_1.x_2.x_3.x_4.x_5 = 1 $.
Prove that \[ (ax_1^2+bx_1+c)(ax_2^2+bx_2+c)...(ax_5^2+bx_5+c)\ge 1\]
Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]
Calculate the following indefinite integrals.
[1] $\int \sin x\cos ^ 3 x dx$
[2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$
[3] $\int x^2 \sqrt{x^3+1}dx$
[4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$
[5] $\int (1-x^2)e^x dx$
The game Prongle is played with a special deck of cards: on each card is a nonempty set of distinct colors. No two cards in the deck contain the exact same set of colors. In this game, a “Prongle” is a set of at least $2$ cards such that each color is on an even number of cards in the set. Let k be the maximum possible number of prongles in a set of $2019$ cards. Find $\lfloor \log 2 (k) \rfloor$.
a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that
\[ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}. \]
b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn) = f(m)+f(n)$ for all $m,n\in \{1,2,\ldots,k\}$ with $mn\leq k$.
[i]Mihai Baluna[/i]
Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$
Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$
Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$
[i]2010 Nara Medical University entrance exam[/i]
The [i]tower function of twos[/i] is defined recursively as follows: $ T(1) \equal{} 2$ and $ T(n \plus{} 1) \equal{} 2^{T(n)}$ for $ n\ge1$. Let $ A \equal{} (T(2009))^{T(2009)}$ and $ B \equal{} (T(2009))^A$. What is the largest integer $ k$ such that
\[ \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}
\]is defined?
$ \textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$
Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.
(1) A function is defined $ f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2})$ for $ x\geq 0$. Find $ f'(x)$.
(2) Find the arc length of the part $ 0\leq \theta \leq \pi$ for the curve defined by the polar equation: $ r \equal{} \theta\ (\theta \geq 0)$.
Remark:
[color=blue]You may not directly use the integral formula of[/color] $ \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2}$ here.
Determine the largest number in the infinite sequence
\[ 1, \sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4}, \dots, \sqrt[n]{n},\dots\]
Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$
If $\log_MN=\log_NM$, $M\ne N$, $MN>0$, $M\ne 1$, $N\ne 1$, then $MN$ equals:
$\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}$
Calculate the following indefinite integrals.
[1] $\int \sin x\sin 2x dx$
[2] $\int \frac{e^{2x}}{e^x-1}dx$
[3] $\int \frac{\tan ^2 x}{\cos ^2 x}dx$
[4] $\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx$
[5] $\int \frac{e^x}{e^x+1}dx$
Solve the system of equations $$
\left\{\begin{array}{l}
x \log x+y \log y+z \log x=0\\ \\
\dfrac{\log x}{x}+\dfrac{\log y}{y}+\dfrac{\log z}{z}=0
\end{array} \right.
$$
prove that $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\pi}{4}$
Answer the following questions:
(1) Let $a$ be non-zero constant. Find $\int x^2 \cos (a\ln x)dx.$
(2) Find the volume of the solid generated by a rotation of the figures enclosed by the curve $y=x\cos (\ln x)$, the $x$-axis and
the lines $x=1,\ x=e^{\frac{\pi}{4}}$ about the $x$-axis.
Let $ p$ be a prime number, $ p\not \equal{} 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 \plus{} b^3$. Prove that $ p^2\mid a \plus{} b$ or $ p^3\mid a^3 \plus{} b^3$.
Consider \[A = \log (2013 + \log (2012 + \log (2011 + \log (\cdots + \log (3 + \log 2) \cdots )))).\] Which of the following intervals contains $ A $?
$ \textbf{(A)} \ (\log 2016, \log 2017) $
$ \textbf{(B)} \ (\log 2017, \log 2018) $
$ \textbf{(C)} \ (\log 2018, \log 2019) $
$ \textbf{(D)} \ (\log 2019, \log 2020) $
$ \textbf{(E)} \ (\log 2020, \log 2021) $
Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left\{\frac{1}{\tan x\ (\ln \sin x)}+\frac{\tan x}{\ln \cos x}\right\}\ dx.$
Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$
Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation:
\[a_{n+1}=10^n a_n^2\]
(a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$.
(b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.