Found problems: 9
Find the infinite integrals as follows.
(1) 2013 Hiroshima City University entrance exam/Informatic Science
$\int \frac{x^2}{2-x^2}dx$
(2) 2013 Kanseigakuin University entrance exam/Science and Technology
$\int x^4\ln x\ dx$
(3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam
$\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$
Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations
$$ \int_{0}^{x} P(t)\sin t \, dt =0, \;\;\;\; \int_{0}^{x} P(t) \cos t \, dt =0 $$
has only finitely many solutions $x.$
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]
Find all differentiable functions $ f:(0,\infty )\longrightarrow (-\infty ,\infty ) $ having the property that
$$ f'(\sqrt{x}) =\frac{1+x+x^2}{1+x} , $$
for any positive real numbers $ x. $
[i]Ovidiu Țâțan[/i]
Let be three positive real numbers $ a,b,c, $ a natural number $ n, $ and the functions $ f:\mathbb{R}\longrightarrow\mathbb{R} ,g:(0,\infty )\longrightarrow\mathbb{R} $ defined as:
$$ f(x)=\frac{2(n+1)x^n(x^{n+1}-a) +nx^{n+1} +2a^2x+a}{x^{2n+2}-2ax^{n+1} +a^2x^2+a^2} , $$
$$ g(x)=\frac{a+bx^n}{x+cx^{2n+1}} $$
Calculate the antiderivatives of $ f $ and $ g. $
[i]Nicolae Sanda[/i]
Let be two real numbers $ a>0,b. $ Calculate the primitive of the function $ 0<x\mapsto\frac{bx-1}{e^{bx}+ax} . $
[i]Dan Negulescu[/i]
$ \int \frac{1+2x^3}{1+x^2-2x^3+x^6} dx $
[i]Ion Nedelcu[/i] and [i]Lucian Tutescu[/i]
Determine the primitives of:
[b]1)[/b] $ (0,\pi /2)\ni x\mapsto\frac{x^2}{-x+\tan x} $
[b]2)[/b] $ 1<x\mapsto \frac{-1+\ln x}{x^2-\ln^2 x} $
[i]Ion Nedelcu[/i]
Find the infinite integrals as follows.
(1) 2013 Hiroshima City University entrance exam/Informatic Science
$\int \frac{x^2}{2-x^2}dx$
(2) 2013 Kanseigakuin University entrance exam/Science and Technology
$\int x^4\ln x\ dx$
(3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam
$\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$