4x
f(x) = ∫ et/t dt , where x ε R+ , THEN THE COMPLETE SET of x for which f (x) ≤ ln x is ??
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58 Answers
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21sir how come all continuous functions are differentiable.. sir i think that it must be all differentiable functions r continous but not all continous functions r differntiable.....
1eg. mod ln x at x=1,
it is continous at x=1 but not differiantiable
62see i was talking about integral functions.. b;ut mind and fingers were not in sync.. i meant integrable (obviously not differentiable.. )
21any function continous on an interval say (a,b) has an antiderivative in that interval......this is to say that there exists a function F(x) such that F'(x) =f(x)....but however not every antiderivative F(x) even when it exist is expressible in the closed form in terms of elemntary functions such as polynomials trigo logarithmic or exponential functs ets....then we say that such antiderivative doenst exist or function is not integrable
62who told the last thing?
but however not every antiderivative F(x) even when it exist is expressible in the closed form in terms of elemntary functions such as polynomials trigo logarithmic or exponential functs ets....then we say that such antiderivative doenst exist or function is not integrable ???
???
62Our syllabus for IIT uses Riemann Integral..
The sum of small areas if you remember that correctly.,.. (http://en.wikipedia.org/wiki/Riemann_integral)
But generally (at a higher level) used is Lebesgue integral
You dont need to get into those complications though...
1check this......
http://targetiit.com/iit-jee-forum/posts/integrate-10513.html
4Copied from the link mentioned by Rahul
It is not correct to say that it is not integrable. Indeed, let
It is easy to see that
so that F(x) is the antiderivative of eX2
The point is that the integral cannot be expressed in terms of the elementary function. But that's not the same as being not integrable.
21fourier series is not really in jee syllab so wats the need to do a q out of syllab....
1all differian. function are continous not all continous function are differiantiable
4ONE MORE this i guess is an interesting one :
∫0x [sint] dt , where x ε (2nΠ, (2n+1)Π) , n ε N & [.] -> greatest integer function is equal to
a) -nÎ b) -(n+1)Î c) -2nÎ d) -(2n+1)Î
3#47 fr no x belonging to R+ the inequality is valid?????
df(x)/dx = e^x/x.....(1)
d lnx /dx = 1/x .......(2)
f(0)-->∞ ln(0)--> -∞ ..........
so the given inequality may be true only if (1/x)>(e^x/x)
or x<0 !!!!!!!
so no set of values!!!!
3#48
THE given fxn is zero in all I &II quad only -1 in III&IV!!!!!
so before 2npi ....... it has gone down n times
so
-npi is da answer!!!
4#47
@iitimcomin
Ur solution is correct but i typed the qs wrong the limits r from 1 to x & not 0 to x
The actual ans is (0,1]
But I wan't the proof since it's a good one!!!
I 'm soooooo sorry for the mistyping
4# 48
@iitimcomin ur ans is correct !!!!
But coming to the solution how can we say that [sint] is 0 in I & II quads
It can also be 1 at pi/2 right???
& " before 2npi ....... it has gone down n times " by this u mean that it has taken -1 n times right ???
Plz explain this
11need not worry about integration of xx.
Won't help u in JEE
They will ask simple and u will try to look the stars .
Remember only a few questions are real tough and most are basic.
Just keep ur basics clear and xx is not upto +2 level or jee level!!
4# 46
derivative of ∫et / t = ex/x
derivative of ln x = 1/x
ex/x > 1/x for a l l x>1
4Time for moving forward :
β + 2 1∫ 0 x2 e - x2 dx = 1∫ 0 e - x2 dx
Then β =
A) e B) 1/e C) e/2 D) e2 / 2
23#46
h(x)=f(x) -logx\leq 0
h'(x)=\frac{e^{x}-1 }{x}
x> 0,hence, e^{x}-1>0 , hence, h'(x)>0
hence h(x) is an increasing function
now\; h(x) = \int_{1}^{x}{\frac{e^{t}}{t}}dt - logx
now\;clearly,h(x)=0\; for\; x=1
hence\; h(x) \leq 0\; for x\in (-infinity, 1]
@ uttara ...in #55 are u saying that
if f'(x) > g'(x)
then f(x) >g(x) ?????????i dont think so it is always true
1GIVEN A f:n "g" continous for every x→R such that g(1)=5 &
∫10 g(t)dt =2; if o∫1 (x-t)2 g(t) d(t) .
find f'''(1) -f''(1) ???