What is $int_1^oo 1/((1+x)(x^(1/2)))dx$ ?

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Answer 1 · 2015-11-09T20:14:48

$int_1^oo 1/((1+x)sqrt(x))dx = 2int_1^oo 1/(2(1+x)sqrt(x))$

$u = sqrt(x)$
$u^2=x$

if $sqrt(x) = oo$ then $u = oo$
if $sqrt(x) = 1$ then $u = 1$

$du = 1/(2sqrt(x))dx$

So now we have

$int_1^oo 1/(1+u^2)du$

$= 2[arctan(u)]_1^oo$

$= 2(arctan(oo) - arctan(1))$

$=2(pi/2-pi/4) = pi/2$

What is int_1^oo 1/((1+x)(x^(1/2)))dxint_1^oo 1/((1+x)(x^(1/2)))dx?