<s>[INST] Change of variable to show $\int_0^{\infty}\frac{dx}{1+x^2}=2\int_0^1\frac{dx}{1+x^2}$ Which change of variable can be used to show
$$\int_0^{\infty}\frac{dx}{1+x^2}=2\int_0^1\frac{dx}{1+x^2}$$
(direct integration isn't allowed, must use change of variable)
 [/INST] $$I=\int_0^\infty\frac{1}{1+x^2}dx=\int_0^1\frac{1}{1+x^2}dx+\int_1^\infty\frac{1}{1+x^2}dx$$
now:
$$I_1=\int_1^\infty\frac{1}{1+x^2}dx$$
$$u=\frac1x,du=-1/x^2dx\Rightarrow dx=-\frac{1}{u^2}du$$
$$I_1=-\int_1^0\frac{1}{1+(1/u)^2}\frac{du}{u^2}=\int_0^1\frac{1}{1+u^2}$$
$$\therefore I=2\int_0^1\frac{1}{1+x^2}dx$$
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