Question

Решить неопределенный интеграл sin(ln x) dx

Answer 1

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

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