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  1. J

    Irrational Integration

    (a)\; $Evaluation of $\int\frac{\sqrt{9x^2+4x+6}}{8x^2+4x+6}dx$ (b)\; $Evaluation of $\int\frac{x^2(x\sec x+\tan x)}{(x\tan x-1)^2}dx.$
  2. J

    floor summ

    $Consider a sequence $b_{n}$ given as $b_{1}=\frac{1}{3}$ $ and $b_{n+1}=b^2_{n}+b_{n}$. Then value of $\lfloor \sum^{2008}_{k=2}\frac{1}{b_{k}}\rfloor.
  3. J

    Definite Integration

    $Evaluation of $\int^{\frac{\pi}{2}}_{0}\cos x\cdot \ln(\cos x)dx
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    HSC 2018-2019 MX2 Integration Marathon

    Re: HSC 2018 MX2 Integration Marathon $Let $I = \int^{1}_{0}\frac{\ln(1+x)}{1+x^2}dx,$ $Put $x=\frac{1-t}{1+t}=\frac{2}{t+1}-1\;,$ Then $dx=-\frac{2}{(1+t)^2}dt $And changing limits$ $So $I = \int^{1}_{0}\frac{\ln(2)-\ln(1+t)}{1+t^2}dt=\int^{1}_{0}\frac{\ln(2)}{1+t^2}dt-I$ $So we have $2I...
  5. J

    Rational Integration

    Evaluation of \displaystyle \int\frac{x^5+3}{(x-5)^3\cdot (x-1)}dx Although i have solved it using partial fraction. But it is very lengthy so please explain me any bettter method. Thanks.
  6. J

    Limit with binomial Coefficients

    To paradoxica would you like to explain me in detail. Thanks
  7. J

    Limit with binomial Coefficients

    $Finding value of $ $\displaystyle \lim_{n\rightarrow \infty} n^{-2}\cdot \sum^{n}_{k=0}\ln\bigg(\binom{n}{k}\bigg)$
  8. J

    Trigonometric product

    $ Finding value of $ (0.5+\cos \alpha)\times(0.5+\cos 3\alpha)\times(0.5+\cos 9\alpha)\times(0.5+\cos 27\alpha)$ $ Where $\alpha =9^\circ}$ I have tried like this way $ let $(x+\cos \alpha)(x+\cos 3\alpha)(x+\cos 9\alpha)(x+\cos 27\alpha)$ $which equals $x^4+(\sum \cos \alpha)x^3+(\cos...
  9. J

    Sequence and series

    Thanks camelrider.
  10. J

    Sequence and series

    $Let $\displaystyle a_{n} = n+\frac{1}{n}$ for $n=1,2,3,4,....20$ and $p=\frac{1}{20}\sum^{20}_{n=1}a_{n}$ $And $q = \frac{1}{20}\sum^{20}_{n=1}\frac{1}{a_{n}}$. Then proving $q\in \left(0,\frac{21-p}{21}\right)$. Trial p = \frac{1}{20}\left((1+2+3+\cdot +20)+1+\frac{1}{2}+\frac{1}{3}+\cdots...
  11. J

    Largest prime number

    $Finding largest prime number which divide $\displaystyle \binom{2000}{1000}$ $We can write $\displaystyle \binom{2000}{1000} = \frac{2000!}{1000!\cdot 1000!} = \frac{1001 \cdot 1002 \cdot 1003\cdots \cdots 2000}{1000!}$ i did not understand how to solve further, please explain me
  12. J

    Probability Question

    $There are four machines and it is known that exactly two of them are $ $faulty. They are tested, one by one, in a random order till both the faulty $ $machines are identified. Then the probability that only two tests are$ $need is$ $What i have Try:$ $Let $A$ be the event in...
  13. J

    integer ordered pair in combination

    $Total number of positive integer ordered pair of $\binom{a}{b} = 120$ $Using $\binom{a}{b} = 120 = \binom{120}{1} = \binom{120}{119}$. So $(a,b) = (120,1)\;,(120,119)$ $And $\binom{a}{b}$ is $\max$, when $b=\frac{a}{2}$ or $b=\frac{a+1}{2}$ $So must have $b\leq \frac{a}{2}$ or $b \leq...
  14. J

    problem involving floor sum

    \displaystyle \bigg\lfloor\frac{2017!}{1!+2!+3!+\cdots \cdots +2016!}\bigg\rfloor $where $\lfloor x \rfloor $ is a floor of $x$ $For upper bound $1!+2!+3!+\cdots \cdots +2016!=2016(1!+2!+3!+\cdots +2016!)$ 2016(1!+2!+3!+\cdots \cdots +2016!)>2016(2016!+2015!)=2017! \displaystyle...
  15. J

    Infinite series sum

    my bad omegadot you are Right., I have got it, Thanks
  16. J

    ratio of two definite Integration

    Sorry friends actually original question as $If $I = \int^{1}_{0}x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}dx$ and $J = \int^{1}_{0}\frac{x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}}{(3+x)^8}dx$ . Then $\displaystyle \frac{I}{J}$ is
  17. J

    Infinite series sum

    $Evaluation of sum of series $\lim_{n\rightarrow \infty}\sum^{n}_{k=1}\frac{1}{(k-1)\sqrt{k}+k\sqrt{k+1}}$
  18. J

    4u trig proof

    $Using $\sin^2 x+\cos^2 x = 1\Rightarrow \sin^2 x+\cos^2 x+(-1) = 0$ $Now if $a+b+c=0\;,$ Then $a^3+b^3+c^3=3abc$ $So $(\sin^2 x)^3+(\cos^2 x)^3+(-1)^3=-3\sin^2 x\cdot \cos^2 x$ $So $\sin^6 x+\cos^6 x= \frac{1}{4}\left[4-3(\sin 2x)^2\right] = \frac{1}{4}\left[1+3\cos^2 (2x)\right].$
  19. J

    ratio of two definite Integration

    $If $I = \int^{1}_{0}x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}dx$ and $J = \int^{1}_{0}\frac{x^{\frac{3}{2}}(1-x)^{\frac{7}{2}}}{(3+x)^8}dx$ . Then $\displaystyle \frac{I}{J}$ is
  20. J

    complex number

    Sorry pikachu97 , i have edited my post.