Double Integral Laws: Moving & Changing Order

[nhrock3]

$$\int_{0}^{\infty}fdx\int_{\frac{x-tx}{t}}^{\infty}dy=\int_{0}^{\infty}dx\int_{\frac{x-tx}{t}}^{\infty}fdy$$

f is a function of x and y

can i move f like i showed?

can i change the order of integration
?

 

[owlpride]

As stated, your integral does not exist because the term

$$\int_{\frac{x-tx}{t}}^{\infty}dy$$

diverges. To answer your question more generally, yes, you may move f provided it is only a function of x and not of y. In that case f is a constant w.r.t. y, and you may move constants in and out of an integral. If f is a function of y, it *has* to be inside the dy integral - your left-hand integral would not make sense. By the way, I am assuming your integral is intended to be

$$\int_{0}^{\infty}f(x) \left( \int_{\frac{x-tx}{t}}^{\infty}dy \right) dx$$