Question

Решить неопределенный интеграл sin(ln x) dx

Answer 1

$intlimits {sin(ln (x))} , dx = intlimits {xsin(ln(x)) frac{1}{x} } , dx = intlimits {xsin(ln(x))} , d(ln(x))=\ = intlimits {e^tsin(t)} , dt = intlimits {sin(t)} , d(e^t)=e^tsin(t)- intlimits {e^t} , d(sin(t)) =\ =e^tsin(t)- intlimits {e^tcos(t)} , dt =e^tsin(t)- intlimits {cos(t)} , d(e^t) =\ =e^tsin(t)- e^tcos(t)+intlimits {e^t} , d(cos(t)) =\ =e^t[sin(t)- cos(t)]-intlimits {e^tsin(t)} , dt =intlimits {e^tsin(t)} , dt$

$=intlimits {e^tsin(t)} , dt=frac{e^t}{2}[sin(t)- cos(t)]+C=\= frac{x}{2}[sin(ln(x))-cos(ln(t))] +C$