I'm working on a problem where I have to calculate the conditional expectation, but I get inconsistent results. I double-checked the integral calculations using Mathematica, so I think my mistake is somewhere in the integrals chosen.
We have the following joint density function: $$f_{X,Y}(x,y)=\begin{cases}\frac{x^y}{\log(2)} &\text{for $x,y\in(0,1)$}\\ 0 &\text{otherwise} \end{cases}\\f_Y(y)=\int_0^1f_{X,Y}(x,y)\mathrm{d}x=\frac{1}{\log(2)(y+1)}\\ \mathbb{E}(X|Y=y)=\int_0^1x\frac{f_{X,Y}(x,y)}{f_Y(y)}\mathrm{d}x=\frac{1}{(y+1)(y+2)}\\ \mathbb{E}(X)=\int_0^1xf_{X,Y}(x,y)\mathrm{d}x\mathrm{d}y=\frac{\log(\frac32)}{\log2}\\ \mathbb{E}(X)=\int_0^1\mathbb{E}(X|Y=y)f_Y(y)\mathrm{d}y=\int_0^1\frac{\mathrm{d}y}{\log(2)(y+1)^2(y+2)}=\frac{\frac12+\log(\frac34)}{\log2}\not=\frac{\log(\frac32)}{\log2}$$
Also, double-checking using the marginal density function $f_X$ seems to confirm $\frac{\log(\frac32)}{\log2}$ should be the right answer.
Which is clearly a contradiction. So what did I do wrong? Any help is much appreciated!