Suppose $x$ and $\lambda$ are real number,
Are there any real-valued function $f(x,\lambda)$ and $g(x)$ satisfying following equation?
$\int f(x,\lambda)dx=\left(\int g(x)dx\right)^\lambda$
General condition to satisfy this equation is more helpful.
Thank you in advance!
That's actually a surprisingly interesting question. Here's a way to get started that seems to lead somewhere: first, for convenience, write $ A(\lambda) = \int {f(x,\lambda)dx} $ and $ G(x) = \int{g(x)dx} $.
If you raise both sides to the $ 1/\lambda $ you get $ A(\lambda)^{1/\lambda} = G(x) $. Now let's differentiate both sides of this with respect to $ \lambda $. Since the right-side doesn't depend on $ \lambda $, we get zero on the right-hand side. The calculations give you $$ A(\lambda)^{1/\lambda} \left(\cfrac{A'(\lambda)}{A(\lambda)\lambda} - \cfrac{\log A(\lambda)}{\lambda^2} \right) = 0 $$ One solution of course is $ A(\lambda) \equiv 0 $, but more interestingly, setting the second factor to zero and clearing denominators gives you a nice, separable differential equation, $$ \lambda A'(\lambda) - A(\lambda) \log(A(\lambda)) = 0 $$ Try going from there.