How do you evaluate the definite integral $int (2x)/(1+x^2)$ from $[0,1]$ ?

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Answer 1 · 2016-08-13T21:19:17

$int_0^1 (2x)/(1+x^2)d x=0.6931471806=l n(2)$

$int_0^1 (2x)/(1+x^2)d x=?$

$"Substitute "u=1+x^2$
$d u=2x*d x$

$int (2x)/(1+x^2)d x=int(d u)/u=l n (u)$

$"Undo substitution"$

$int_0^1 (2x)/(1+x^2)d x=|l n(1+x^2)|_0^1$

$int_0^1 (2x)/(1+x^2)d x=[l n(1+1^2]-[1+0^2]$

$int_0^1 (2x)/(1+x^2)d x=[l n (2)]-[l n(1)]$

$l n(2)=0.6931471806$
$l n(1)=0$

$int_0^1 (2x)/(1+x^2)d x=0.6931471806-0$

$int_0^1 (2x)/(1+x^2)d x=0.6931471806$

How do you evaluate the definite integral int (2x)/(1+x^2)int (2x)/(1+x^2) from [0,1][0,1]?