Question

Интеграл cos(Inx)dx помогите, пожалуйста.

Answer 1

$\large \\ \int\cos{(\ln{x})}\mathrm{dx}=\begin{vmatrix} u=\cos{(\ln{x})}, du=-\sin{(\ln{x})}\cdot{1\over x}\mathrm{dx}\\ \mathrm{dv}=\mathrm{dx}, v=x \end{vmatrix}=x\cdot\cos{(\ln{x})}-(-1)\cdot\int\sin{(\ln{x})}\mathrm{dx}=\begin{vmatrix} u=\sin{(\ln{x})}, du=\cos{(\ln{x})}\cdot{1\over x}\mathrm{dx}\\ \mathrm{dv}=\mathrm{dx}, v=x \end{vmatrix}=x\cdot\cos{(\ln{x})}+x\cdot\sin{(\ln{x})}-\boldsymbol{\int\cos{(\ln{x})}\mathrm{dx}}=*\\ I=\int\cos{(\ln{x})}\mathrm{dx}\\ I=x\cdot\cos{(\ln{x})}+x\cdot\sin{(\ln{x})}-I\\ 2I=x\cdot\cos{(\ln{x})}+x\cdot\sin{(\ln{x})}\\ I={x\over2}\cdot\begin{pmatrix} \cos{(\ln{x})}+\sin{(\ln{x})} \end{pmatrix}\\\\ *={x\over2}\cdot\begin{pmatrix} \cos{(\ln{x})}+\sin{(\ln{x})} \end{pmatrix}$