Found problems: 713
2007 Today's Calculation Of Integral, 206
Calculate $\int \frac{x^{3}}{(x-1)^{3}(x-2)}\ dx$
2005 Today's Calculation Of Integral, 47
Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$.
\[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]
1978 Putnam, A3
Find the value of $ k\ (0<k<5)$ such that $ \int_0^{\infty} \frac{x^k}{2\plus{}4x\plus{}3x^2\plus{}5x^3\plus{}3x^4\plus{}4x^5\plus{}2x^6}\ dx$ is minimal.
2009 Today's Calculation Of Integral, 481
For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$.
Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$
2010 Today's Calculation Of Integral, 661
Consider a sequence $1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots$.
(1) Find the 36th term.
(2) Find $\int x^2\ln x\ dx$.
(3) Let $A$ be the product of from the first term to the 36th term. How many digits does $A$ have integer part?
If necessary, you may use the fact $2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31$.
[i]2010 National Defense Medical College Entrance Exam, Problem 4[/i]
2010 Today's Calculation Of Integral, 546
Find the minimum value of $ \int_0^{\pi} \left(x \minus{} \pi a \minus{} \frac {b}{\pi}\cos x\right)^2dx$.
2007 Today's Calculation Of Integral, 235
Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.
2011 Today's Calculation Of Integral, 693
Evaluate $\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.$
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2005 Today's Calculation Of Integral, 88
A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$
Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$
2012 Today's Calculation Of Integral, 834
Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$.
2007 Today's Calculation Of Integral, 250
For a positive constant number $ p$, find $ \lim_{n\to\infty} \frac {1}{n^{p \plus{} 1}}\sum_{k \equal{} 0}^{n \minus{} 1} \int_{2k\pi}^{(2k \plus{} 1)\pi} x^p\sin ^ 3 x\cos ^ 2x\ dx.$
2007 Today's Calculation Of Integral, 221
Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.
2009 Today's Calculation Of Integral, 443
Evaluate $ \int_1^{e^2} \frac{(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)\plus{}(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)}{\sqrt{x}}\ dx.$
2009 Today's Calculation Of Integral, 435
Evaluate $ \int_{\frac{\pi}{4}}^{\frac {\pi}{2}} \frac {1}{(\sin x \plus{} \cos x \plus{} 2\sqrt {\sin x\cos x})\sqrt {\sin x\cos x}}dx$.
2009 Today's Calculation Of Integral, 428
Let $ f(x)$ be a polynomial and $ C$ be a real number.
Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.
2009 Today's Calculation Of Integral, 414
Evaluate $ \int_0^{2(2\plus{}\sqrt{3})} \frac{16}{(x^2\plus{}4)^2}\ dx$.
Today's calculation of integrals, 881
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.
2009 Today's Calculation Of Integral, 419
In the $ xy$ plane, the line $ l$ touches to 2 parabolas $ y\equal{}x^2\plus{}ax,\ y\equal{}x^2\minus{}2ax$, where $ a$ is positive constant.
(1) Find the equation of $ l$.
(2) Find the area $ S$ bounded by the parabolas and the tangent line $ l$.
2007 Today's Calculation Of Integral, 168
Prove that $\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}{\left|\frac{1}{x}\sin \frac{\pi}{x}\right| dx}$ diverge for $x>0.$
2010 Today's Calculation Of Integral, 626
Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$
[i]2010 Nara Medical University entrance exam[/i]
2007 ISI B.Stat Entrance Exam, 2
Use calculus to find the behaviour of the function
\[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\]
and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.
2011 Today's Calculation Of Integral, 716
Prove that :
\[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]
2010 Today's Calculation Of Integral, 608
For $a>0$, find the minimum value of $\int_0^1 \frac{ax^2+(a^2+2a)x+2a^2-2a+4}{(x+a)(x+2)}dx.$
2010 Gakusyuin University entrance exam/Science
2010 Today's Calculation Of Integral, 641
Evaluate
\[\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.\]
Own
2009 Today's Calculation Of Integral, 478
Evaluate $ \int_0^{\frac{\pi}{4}} \{(x\sqrt{\sin x}\plus{}2\sqrt{\cos x})\sqrt{\tan x}\plus{}(x\sqrt{\cos x}\minus{}2\sqrt{\sin x})\sqrt{\cot x}\}\ dx.$