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diff --git a/calc/sol0210.tex b/calc/sol0210.tex new file mode 100644 index 0000000..bb4ad94 --- /dev/null +++ b/calc/sol0210.tex @@ -0,0 +1,56 @@ +\documentclass[10pt,a5paper]{article} +\usepackage[svgnames, rgb]{xcolor} + +\input{intro} + +\lhead{\color{gray} Шарафатдинов Камиль 192} +\rhead{\color{gray} \texttt{sol0203}} +\title{ДЗ на 10.02} +\author{Шарафатдинов Камиль БПМИ-192} +\date{билд: \today} + + +% -- Here bet dragons -- +\begin{document}\thispagestyle{empty} + +\maketitle +\clearpage +\setcounter{page}{1} + +\question{1.f}{ + \[ + \int_1^e \sin \log x \dif x + \] +} + + \begingroup + \setlength{\jot}{8pt} + \begin{align*} + I = \int \sin \log \dif x + &= \int e^u \sin u \dif u + &\explain{ + \displaystyle u = \log x\\ + \displaystyle \dif u = \frac{\dif x}{x} = \frac{\dif x}{e^u} + }\\ + &= - e^u \cos u + \int e^u \cos u \dif u\\ + &= - e^u \cos u + e^u \sin u - \int e^u \sin u \dif u + \end{align*} + \endgroup + + \begin{flalign*} + 2I = e^u (\sin u - \cos u) + C\\ + I = \frac{e^u}{2} (\sin u - \cos u) + C = + \frac{x}{2} (\sin \log x - \cos \log x) + C + \end{flalign*} + + \[ + \int_1^e \sin \log x \dif x = + \frac{e}{2} (\sin 1 - \cos 1) - \frac{1}{2} (0 - 1) = + \frac{e}{2} (\sin 1 - \cos 1) + \frac{1}{2} + \] + +\vspace*{\fill} + + P.S. Ну проверь хотя бы одну задачу, пожаааалуйста + +\end{document} |