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## Homework Statement

Prove the following inequality:

[tex]\frac{1}{6}\leq\int_{R}\frac{1}{y^{2}+x+1}\chi_{B}(x,y)dxdy\leq\frac{1}{2}[/tex]

where B={(x,y)|0[tex]\leq (x)\leq (y)\leq1[/tex]} and R=[0,1]x[0,1]

EDIT: The B region should be 0 less than or equal to x less than or equal to y less than or equal to 1.

## Homework Equations

I understand the [tex]\chi_{B}[/tex] to be the characteristic function, ie takes value 1 if x is in B, and zero else.

## The Attempt at a Solution

I've bashed my head against a wall for ages with this one. It keeps coming out wrong and different every time.

I don't know if whether to solve it is to brute force integrate on the double integral the function like so:

[tex]\int^{1}_{0}\int^{y}_{0}\frac{1}{y^{2}+x+1}dxdy[/tex] (or equivalently the second integral between x and 1, and do dydx, which i believe is the same thing.)

Is this the right thing to do? Or is there a much quicker, shortcut way of doing it?

Many thanks,