I met the following question and I don't know how to compute it directly:
Let $G:=GL(2,\mathbb{R})$, and equip $G$ with the Haar measure $$dg:=\frac{1}{|det(g)|^2}dg_{11}...dg_{22},~(g:=(g_{ij})).$$ Pick the following function defined on $G$: $$f(g):=exp(-(g_{11}^2+g_{12}^2+g_{21}^2+g_{22}^2)).$$ I want to compute $\int_G f(g) dg$, but I have no idea how to reduce this integral to a "usual" one (in multi-variable calculus). Can anyone help me or give me some hint? Thanks a lot in advance!
(p.s. The background for such a calculation is related to the evaluation of some classical automorphic $L$-functions at archimedean primes. I found that using the measure obtained from the Iwasawa decomposition would be easier. But another problem appears: the two measures differ by a constant, and I don't know the constant yet.)
