# Search results

1. ### Irrational Integration

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

9. ### Sequence and series

Thanks camelrider.
10. ### 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. ### 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

14. ### problem involving floor sum

\displaystyle \bigg\lfloor\frac{2017!}{1!+2!+3!+\cdots \cdots +2016!}\bigg\rfloor $where$\lfloor x \rfloor $is a floor of$xFor 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. ### Infinite series sum

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

18. ### 4u trig proof

$Using$\sin^2 x+\cos^2 x = 1\Rightarrow \sin^2 x+\cos^2 x+(-1) = 0Now 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 xSo $\sin^6 x+\cos^6 x= \frac{1}{4}\left[4-3(\sin 2x)^2\right] = \frac{1}{4}\left[1+3\cos^2 (2x)\right].$
19. ### 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. ### complex number

Sorry pikachu97 , i have edited my post.