Question

anti derivative of ln(1+x)/(1+x^2)​

Answer 1

Answer:

$log_{e} \frac{1 + x}{1 + {x}^{2} } \\ \\ we \: know \: that \: log_{e} = 1 \\ \\ take \: f(x) = \frac{1 + x}{1 + {x}^{2} } \\ \\ f'(x) = \frac{( 1 + {x}^{2} ) \frac{d}{dx} (1 + x) - (1 + x) \frac{d}{dx} (1 + {x}^{2} )}{(1 + {x}^{2}) ^{2} } \\ \\ \: \: \: \: \: \: \: \: \: = \frac{(1 + {x})^{2}(1) - (1 + x)(2x) }{(1 + {x}^{2} ) ^{2} } \\ \\ \: \: \: \: \: \: \: \: = \frac{(1 + x ^{2} ) - 2x - 2 {x}^{2} }{(1 + {x}^{2} ) ^{2} } \\ \\ \: \: \: \: \: \: \: \: = \frac{1 - {x}^{2} - 2x }{ (1 + {x}^{2} ) ^{2} } \\ \\ \: \: \: \: \: \: \: \: \: \:$