Question

Найдите неопределённый интеграл:
$\int\limits {sin(lnx)} \, dx$

Answer 1

$\displaystyle \int \sin(\ln x)\, dx= \left\{\begin{array}{ccc}t=\ln x\\ x=e^t\\dt=\dfrac{dx}{x}\end{array}\right\}=\int e^{t}\sin t\, dt\,\,\,\boxed{=}$

$\displaystyle \int e^{\alpha t}\sin(\beta t)dt=\dfrac{e^{\alpha t}(\alpha\sin(\beta t)-\beta \cos(\beta t))}{\alpha^2+\beta^2}+C$

$\boxed{=}~\dfrac{1}{2}e^t(\sin t-\cos t)+C=\dfrac{1}{2}e^t\cdot \sqrt{2}\sin(t-\frac{\pi}{4})=\dfrac{1}{2}e^{\ln x}\sqrt{2}\sin(\ln x-\frac{\pi}{4})+C=\\ \\ \\ =\dfrac{x}{\sqrt{2}}\sin(\ln x-\frac{\pi}{4})+C$