This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 713

2010 Today's Calculation Of Integral, 643
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.\] Own

2005 Today's Calculation Of Integral, 19
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int \tan ^ 3 x dx$ [2] $\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)$ [3] $\int \cos ^ 5 x dx$ [4] $\int \sin ^ 2 x\cos ^ 3 x dx$ [5]$ \int \frac{dx}{\sin x}$

2009 Today's Calculation Of Integral, 469
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt$.

2005 Today's Calculation Of Integral, 86
Tags: calculus , integration , trigonometry , calculus computations
Prove \[\left[\int_\pi^\infty \frac{\cos x}{x}\ dx\right]^2< \frac{1}{{\pi}^2}\]

2005 Today's Calculation Of Integral, 70
Tags: calculus , integration , trigonometry , function , calculus computations
Find the number of root for $\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0$ at $0\leq a <2\pi$

2011 Today's Calculation Of Integral, 729
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\int_1^e \frac{\ln x-1}{x^2-(\ln x)^2}dx.$

2007 Today's Calculation Of Integral, 180
Tags: calculus , integration , geometry , trigonometry , limit , calculus computations
Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

2012 Today's Calculation Of Integral, 815
Tags: calculus , integration , trigonometry , function , calculus computations
Prove that : $\left|\sum_{i=0}^n \left(1-\pi \sin \frac{i\pi}{4n}\cos \frac{i\pi}{4n}\right)\right|<1.$

2010 Today's Calculation Of Integral, 542
Tags: calculus , integration , function , calculus computations
Find continuous functions $ f(x),\ g(x)$ which takes positive value for any real number $ x$, satisfying $ g(x)\equal{}\int_0^x f(t)\ dt$ and $ \{f(x)\}^2\minus{}\{g(x)\}^2\equal{}1$.

2009 Today's Calculation Of Integral, 462
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.

2010 Today's Calculation Of Integral, 545
Tags: calculus , integration , function , calculus computations
(1) Evaluate $ \int_0^1 xe^{x^2}dx$. (2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.

2010 Today's Calculation Of Integral, 574
Tags: calculus , integration , calculus computations , logarithm
Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,

2012 Today's Calculation Of Integral, 795
Tags: calculus , integration , trigonometry , calculus computations , definite integral , logarithm
Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$

2012 Today's Calculation Of Integral, 806
Tags: calculus , integration , trigonometry , calculus computations
Let $n$ be positive integers and $t$ be a positive real number. Evaluate $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$

2009 Today's Calculation Of Integral, 466
Tags: calculus , integration , limit , inequalities , calculus computations , logarithm
For $ n \equal{} 1,\ 2,\ 3,\ \cdots$, let $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$ be the point of intersection of $ y \equal{} \ln (nx)$ and $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$. (1) Show that $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$ to find $ \lim_{n\to\infty} q_n$. (2) Find $ \lim_{n\to\infty} n\int_{\frac {1}{n}}^{p_n} \ln (nx)\ dx$.

2012 Today's Calculation Of Integral, 789
Tags: calculus , integration , function , calculus computations
Find the non-constant function $f(x)$ such that $f(x)=x^2-\int_0^1 (f(t)+x)^2dt.$

2013 Today's Calculation Of Integral, 886
Tags: calculus , integration , function , trigonometry , calculus computations
Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

2007 Today's Calculation Of Integral, 256
Tags: calculus , integration , trigonometry , calculus computations
Find the value of $ a$ for which $ \int_0^{\pi} \{ax(\pi ^ 2 \minus{} x^2) \minus{} \sin x\}^2dx$ is minimized.

2012 Today's Calculation Of Integral, 816
Tags: calculus , integration , geometric transformation , rotation , combinatorial geometry , calculus computations , geometry
Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.

2009 Today's Calculation Of Integral, 504
Tags: calculus , integration , geometry , parabola , calculus computations , conic
Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.

2009 Today's Calculation Of Integral, 489
Tags: calculus , integration , limit , function , calculus computations
Find the following limit. $ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.

2005 Today's Calculation Of Integral, 84
Tags: calculus , integration , limit , trigonometry , calculus computations
Evaluate \[\lim_{n\to\infty} n\int_0^\pi e^{-nx} \sin ^ 2 nx\ dx\]

2008 Harvard-MIT Mathematics Tournament, 7
Tags: geometry , 3d geometry , limit , function , calculus , derivative , calculus computations
([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.

2013 Today's Calculation Of Integral, 878
Tags: calculus , integration , function , trigonometry , calculus computations
A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

2013 Today's Calculation Of Integral, 879
Tags: calculus , integration , trigonometry , calculus computations , integral
Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$