- #1
whoareyou
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Homework Statement
Here is a more interesting problem to consider. We want to evaluate the improper integral
[tex]\intop_{0}^{\infty}\frac{\tan^{-1}(6x)-\tan^{-1}(2x)}{x}dx[/tex]
Do it by rewriting the numerator of the integrand as [tex]\intop_{f(x)}^{g(x)}h(y)dy[/tex] for appropriate [itex]f, g, h[/itex] and then reversing the order of integration in the resulting double integral.
Homework Equations
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The Attempt at a Solution
Writing the numerator of the integrand as an integral:
[tex]\Rightarrow\intop_{0}^{\infty}\frac{\intop_{2x}^{6x}\frac{1}{1+y^{2}}dy}{x}dx[/tex]
[tex]\Rightarrow\intop_{0}^{\infty}\frac{1}{x}\intop_{2x}^{6x}\frac{1}{1+y^{2}}dydx[/tex]
[tex]\Rightarrow\intop_{2x}^{6x}\frac{1}{x}\intop_{0}^{\infty}\frac{1}{1+y^{2}}dxdy[/tex]
But it seems to be a dead end from here since you have to evaluate x at infinity and 0. I don't know how to continue from here.