Olympiad problems

2006 Petru Moroșan-Trident, 2

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]

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: differentiability , primitivableness , antiderivative , integral , indefinite integral , real analysis , function

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]

2013 Bogdan Stan, 3

Tags: indefinite integral , antiderivative , calculus , integration

$ \int \frac{1+2x^3}{1+x^2-2x^3+x^6} dx $ [i]Ion Nedelcu[/i] and [i]Lucian Tutescu[/i]

2006 Petru Moroșan-Trident, 3

Tags: primitive , indefinite integral , calculus

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]

1980 Putnam, A5

Tags: indefinite integral , polynomial , trigonometry , integration

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 $x.$

2007 Mathematics for Its Sake, 3

Tags: integration , primitive , indefinite integral , function

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]

Today's calculation of integrals, 885

Tags: logarithm , integration , indefinite integral , trigonometry , calculus computations , calculus

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 $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

2013 Today's Calculation Of Integral, 885

Tags: calculus , calculus computations , integration , indefinite integral , logarithm , trigonometry

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 $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

2018 Ramnicean Hope, 2

Tags: integration , calculus , function , real analysis , indefinite integral

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]