Plz evaluate :

Rates assured

I didn't understand what u are saying about fractional part . But the formula which u meant , I thinl is -

taking 100 down and writing 1 in place of 100 then what about 10.

Also , one more similar question is:-

In this {x} is fractional part so we can write  {x} as (x-[x]),  But how

can u explain me plz .

I very confused . HELP

A Number   =  It's Integer Part     +     It's Fractional Part

(x)                        [x]                                   {x}

Thus x-[x] = {x}  ,which is the fractional part of x.

The range of {x} is [0,1) and it is periodic (having period = 1).

Thus   is the required integral.

The graph kinda looks like this..............(Note: the graph is for 0<x<5.  Its just the same everywhere else.)

Since its the same for all    n-1< x <n , (where n is an integer)

to find the value of   we must find the area under each of those line segments starting from x = 10 and ending at x = 100.

So,effectively you've got to find the area under those 90 "triangles".

Each "triangle" has an area of  =

So the answer is  = 45

Or you can use the result on the integrals of periodic functions which says that if F(x) is a periodic function with period "n" then

In this case f(x) = {x} and n=1 ....a=10 and b=100.

Thus the reqd integral =

 =

 = 45

Note that you can't integrate it directly because it's discontinous at all xN

and hence for x =10,11,12,...........100

So you have to split it at all it's discontinuities.

i.e.

But since each individual term on the RHS =1/2,and since there are 90 terms,you get the answer as 45.

Coming to your 2nd question i.e. ,

Write {x}= x-[x] and again split the integral at it's discontinuities in (1,4) viz.   x = 2,3

Hence the required integral is

I =

for 1<x<2      [x]=1

for  2<x<3     [x]=2

for  3<x<4     [x]=3

I =

Integrating,we get,

I =    +     +  

I =   +    +

Hence,

I =