# Thread: Need help with algebra within these integrals

1. ## Need help with algebra within these integrals

I'm working on some integrals of exponential families, but I'm just not sure how the textbook have the followings:

1. $\frac {1} {2 \pi } \int \int \exp ( \eta xy - \frac {(x^2+y^2)}{2} ) dxdy$

$= \frac {1}{ \sqrt {2 \pi } } \int \exp [ - \frac {1}{2} (1- \eta ^2)y^2 ] dy$

How does the first line become the second line?

2. Let $A( \eta ) = \log \int ^1 _0 \exp [ \eta \log (x) ] (1-x)^2 dx$

The text says $e ^ { A ( \eta ) } \leq \infty$ only for $\eta \in (-1, \infty )$

Why is that?

2. ## Re: Need help with algebra within these integrals

I'm working on some integrals of exponential families, but I'm just not sure how the textbook have the followings:

1. $\frac {1} {2 \pi } \int \int \exp ( \eta xy - \frac {(x^2+y^2)}{2} ) dxdy$

$= \frac {1}{ \sqrt {2 \pi } } \int \exp [ - \frac {1}{2} (1- \eta ^2)y^2 ] dy$

How does the first line become the second line?

2. Let $A( \eta ) = \log \int ^1 _0 \exp [ \eta \log (x) ] (1-x)^2 dx$

The text says $e ^ { A ( \eta ) } \leq \infty$ only for $\eta \in (-1, \infty )$

Why is that?
1. Integrate wrt x (simple substitution technique - Complete the square, make the substitution, recognise a standard definite integral).

2. Convergence is required.