Olympiad problems

2005 Today's Calculation Of Integral, 41

Tags: calculus , calculus computations , derivative , geometry , trigonometry , integration

Evaluate \[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]

2010 Today's Calculation Of Integral, 582

Tags: inequalities , integration , calculus , trigonometry , calculus computations

Prove the following inequality. \[ \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi\]

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 $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$

2012 Today's Calculation Of Integral, 795

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

Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$

2012 Waseda University Entrance Examination, 4

Tags: calculus , analytic geometry , function , integration , logarithm , calculus computations

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 $Q$.

2009 Today's Calculation Of Integral, 437

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$

Today's calculation of integrals, 856

Tags: geometry , analytic geometry , trigonometry , integration , calculus , calculus computations

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 $a>0$.

2010 Today's Calculation Of Integral, 558

Tags: function , calculus , inequalities , algebra , integration , quadratic , calculus computations

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

Tags: calculus , geometry , integration , calculus computations

Evaluate $\int_{\frac 12}^1 \sqrt{1-x^2}\ dx.$

2007 Today's Calculation Of Integral, 247

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_{\frac{\pi}{8}}^{\frac{3}{8}\pi} \frac{11\plus{}4\cos 2x \plus{}\cos 4x}{1\minus{}\cos 4x}\ dx.$

2012 Today's Calculation Of Integral, 805

Tags: calculus , trigonometry , calculus computations , inequalities , integration

Prove the following inequalities: (1) For $0\leq x\leq 1$, \[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\] (2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$

Today's calculation of integrals, 892

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

Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\cos x}\ dx.$

2011 Today's Calculation Of Integral, 719

Tags: calculus computations , calculus , integration , trigonometry

Compute $\int_0^x \sin t\cos t\sin (2\pi\cos t)\ dt$.

2010 Today's Calculation Of Integral, 596

Tags: trigonometry , calculus computations , calculus , integration

Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$ 2009 Shimane University entrance exam/Medicine

2012 Today's Calculation Of Integral, 845

Tags: calculus computations , integration , calculus

Consider for a real number $t>1$, $I(t)=\int_{-4}^{4t-4} (x-4)\sqrt{x+4}\ dx.$ Find the minimum value of $I(t)\ (t>1).$

2012 Today's Calculation Of Integral, 785

Tags: calculus computations , calculus , integration , derivative , logarithm

For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$

2005 Today's Calculation Of Integral, 9

Tags: logarithm , calculus , absolute value , integration , trigonometry , calculus computations

Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

2012 Today's Calculation Of Integral, 809

Tags: integration , conic , calculus , geometry , parabola , calculus computations

For $a>0$, denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$. Find the maximum area of $S(a)$.

2009 Today's Calculation Of Integral, 456

Tags: integration , logarithm , limit , calculus , trigonometry , real analysis , calculus computations

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\}$

2011 Today's Calculation Of Integral, 731

Tags: integration , calculus , parabola , geometry , conic , calculus computations

Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively. When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.

2005 Today's Calculation Of Integral, 62

Tags: integration , limit , calculus computations , calculus , logarithm

For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$. Evaluate \[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\]

2007 Today's Calculation Of Integral, 217

Tags: integration , calculus , trigonometry , function , geometry , calculus computations , logarithm

Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.

2005 Today's Calculation Of Integral, 36

Tags: calculus , integration , limit , algebra , polynomial , calculus computations

A sequence of polynomial $f_n(x)\ (n=0,1,2,\cdots)$ satisfies $f_0(x)=2,f_1(x)=x$, \[f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)\] Let $x_n\ (n\geqq 2)$ be the maximum real root of the equation $f_n(x)=0\ (|x|\leqq 2)$ Evaluate \[\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx\]

2012 Today's Calculation Of Integral, 776

Tags: calculus computations , calculus , integration

Evaluate $\int_{\frac{1-\sqrt{5}}{2}}^{\frac{1+\sqrt{5}}{2}} (2x^2-1)e^{2x}dx.$

2010 Today's Calculation Of Integral, 603

Tags: derivative , calculus , integration , calculus computations

Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$. Please solve the problem without using partial differentiation for those who don't learn it. 1961 Waseda University entrance exam/Science and Technology