Found problems: 1687
Today's calculation of integrals, 893
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$
2005 Today's Calculation Of Integral, 76
The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows.
\[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\]
Evaluate
\[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]
1998 USAMTS Problems, 5
The figure on the right shows the ellipse $\frac{(x-19)^2}{19}+\frac{(x-98)^2}{98}=1998$.
Let $R_1,R_2,R_3,$ and $R_4$ denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of $R_1-R_2+R_3-R_4$.
[asy]
defaultpen(linewidth(0.7));
pair c=(19,98);
real dist = 30;
real a = sqrt(1998*19),b=sqrt(1998*98);
xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow);
yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow);
draw(ellipse(c,a,b));
label("$R_1$",(100,200));
label("$R_2$",(-80,200));
label("$R_3$",(-60,-150));
label("$R_4$",(70,-150));[/asy]
1983 Miklós Schweitzer, 7
Prove that if the function $ f : \mathbb{R}^2 \rightarrow [0,1]$ is continuous and its average on every circle of radius $ 1$ equals the function value at the center of the circle, then $ f$ is constant.
[i]V. Totik[/i]
2011 AMC 10, 24
A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$?
$ \textbf{(A)}\ \frac{51}{101} \qquad
\textbf{(B)}\ \frac{50}{99} \qquad
\textbf{(C)}\ \frac{51}{100} \qquad
\textbf{(D)}\ \frac{52}{101} \qquad
\textbf{(E)}\ \frac{13}{25} $
2005 Today's Calculation Of Integral, 41
Evaluate
\[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]
2021 CMIMC Integration Bee, 6
$$\int_0^{20\pi}|x\sin(x)|\,dx$$
[i]Proposed by Connor Gordon[/i]
2010 Today's Calculation Of Integral, 582
Prove the following inequality.
\[ \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi\]
2004 Federal Competition For Advanced Students, Part 1, 4
Each of the $2N = 2004$ real numbers $x_1, x_2, \ldots , x_{2004}$ equals either $\sqrt 2 -1 $ or $\sqrt 2 +1$. Can the sum $\sum_{k=1}^N x_{2k-1}x_2k$ take the value $2004$? Which integral values can this sum take?
2012 Today's Calculation Of Integral, 806
Let $n$ be positive integers and $t$ be a positive real number.
Evaluate $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$
2003 Romania Team Selection Test, 5
Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials.
[i]Mihai Piticari[/i]
1980 Putnam, A3
Evaluate
$$\int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.$$
2012 Today's Calculation Of Integral, 795
Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$
2012 Waseda University Entrance Examination, 4
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions:
(1) Find $f'(x)$.
(2) Sketch the graph of $y=f(x)$.
(3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 5
Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers.
Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$
2007 Princeton University Math Competition, 8
For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?
2009 Today's Calculation Of Integral, 437
Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$
2014 Miklós Schweitzer, 9
Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula
\[\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.\]
Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.
Today's calculation of integrals, 856
On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.
2021 CMIMC Integration Bee, 13
$$\int_0^1 x\ln(x^2)\ln(1+x)\,dx$$
[i]Proposed by Connor Gordon[/i]
2010 Today's Calculation Of Integral, 560
Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$.
(1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis.
(2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis.
Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.
1989 IMO Longlists, 27
Let $ L$ denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if one considers four points of $ L$ instead of three?
2010 Today's Calculation Of Integral, 558
For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$.
Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$
2012 Today's Calculation Of Integral, 839
Evaluate $\int_{\frac 12}^1 \sqrt{1-x^2}\ dx.$
2024 CMIMC Integration Bee, 1
\[\int_1^e \frac{\log(x^{2024})}{x} \mathrm dx\]
[i]Proposed by Connor Gordon[/i]