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+\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}
+
+
+\newcommand{\deft}{\texttt{\\deft is undefined}}
+
+
+\question{1.b}{
+ \[
+ \int \frac{\dif x}{\sqrt[6]{(x - 7)^7(x - 5)^5}}
+ = -3\sqrt[6]{\dfrac{x - 5}{x - 7}} + C
+ \]
+}
+
+ \begin{align*}
+ \int \frac{\dif x}{\sqrt[6]{(x - 7)^7(x - 5)^5}}
+ &= -6 \int \frac{t^7 \dif t}{t^7 \br{ \sqrt[6]{x - 5} }^5}
+ &\explain{
+ \displaystyle t = \sqrt[6]{\frac{1}{x - 7}}\\[8pt]
+ \displaystyle \frac{\dif t}{\dif x}
+ = -\frac{1}{6} \br{ \sqrt[6]{\frac{1}{x - 7}} }^7
+ = -\frac{1}{6} t^7
+ }\\[8pt]
+ &= -6 \int \frac{\dif t}{\br{ \sqrt[6]{x - 5} }^5}\\[8pt]
+ &= -6 \int \frac{t^5 \dif t}{\br{ \sqrt[6]{1 + 2t^6} }^5}
+ &\explain{
+ \dfrac{1}{\sqrt[6]{x - 5}}
+ &= \displaystyle \sqrt[6]{\frac{1}{\frac{1}{t^6} + 2}}\\[8pt]
+ &= \dfrac{t}{\sqrt[6]{1 + 2t^6}}
+ }\\[8pt]
+ &= 3 \int \frac{u^5 \dif u}{u^7}
+ = 3 \int \frac{\dif u}{u^2}
+ &\explain{
+ \displaystyle u = \dfrac{1}{\sqrt[6]{1 + 2t^6}}\\
+ \displaystyle \frac{\dif u}{\dif t} = -2t^5 u^7\\
+ }\\
+ &= -\frac{3}{u} + C\\
+ &= -3\sqrt[6]{1 + 2t^6} + C\\
+ &= -3\sqrt[6]{1 + \dfrac{2}{x - 7}} + C\\
+ &= -3\sqrt[6]{\dfrac{x - 5}{x - 7}} + C
+ \end{align*}
+
+\clearpage
+
+\question{7.c}{
+ \[
+ \int \frac{\dif x}{1 + \sqrt{1 - 2x - x^2}}
+ \]
+}
+
+ Воспользуемся почти подстановкой Эйлера:
+ $
+ \displaystyle \sqrt{ax^2 + bx + c} = xt - \sqrt{c}
+ $
+
+ \begin{align*}
+ \int \frac{\dif x}{1 + \sqrt{1 - 2x - x^2}}
+ &= \int \frac{\dif x}{xt}
+ &\explain{
+ \displaystyle \sqrt{1 - 2x - x^2} = xt - 1\\[4pt]
+ \displaystyle 1 - 2x - x^2 = x^2t^2 - 2xt + 1\\[4pt]
+ \displaystyle x \br{ t^2 + 1 } = t - 1\\[4pt]
+ \displaystyle x = \dfrac{t - 1}{t^2 + 1}\\[16pt]
+ \displaystyle \dfrac{\dif x}{\dif t} = -2 \dfrac{t^2 - 2t - 1}{(t^2 + 1)^2}
+ }\\
+ &= \int \frac{-2\dfrac{t^2 - 2t - 1}{(t^2 + 1)^2} \dif t}
+ {2\dfrac{t - 1}{t^2 + 1} t}\\[8pt]
+ &= -\int \frac{(t^2 - 2t - 1)(t^2 + 1) \dif t}
+ {(t - 1)(t^2 + 1)^2 t}\\[8pt]
+ &= -\int \frac{(t^2 - 2t - 1) \dif t}
+ {(t - 1)(t^2 + 1) t}\\[8pt]
+ &= -\int \br{
+ \frac{2}{t^2 + 1} + \frac{1}{t} - \frac{1}{t - 1}
+ } \dif t\\[8pt]
+ &= -2\arctan t - \log{t} + \log(t - 1) + C\\[8pt]
+ &= -2\arctan \dfrac{\sqrt{1 - 2x - x^2} + 1}{x} + \log \dfrac{t - 1}{t} + C\\[8pt]
+ &= -2\arctan \dfrac{\sqrt{1 - 2x - x^2} + 1}{x}
+ + \log \dfrac{\sqrt{1 - 2x - x^2} + 1 - x}{\sqrt{1 - 2x - x^2} + 1} + C\\[8pt]
+ \end{align*}
+
+\clearpage
+
+\question{10.b}{
+ \[
+ \int \frac{x^8 \dif x}{\sqrt{x^2 + 1}}
+ \]
+}
+
+ Лемма (вообще говоря, это задача 9):
+ \begin{gather*}
+ P \in \mathbb{R}_n[x], \qquad
+ Q \in \mathbb{R}_{n - 1}[x], \qquad
+ R = \sqrt{ax^2 + bx + c} \implies
+ \int \frac{P \dif x}{R} = Q R + \lambda \int \frac{\dif x}{R}\\[16pt]
+ \br{ QR + \lambda \int \frac{\dif x}{R}}' =
+ Q'R + QR' + \frac{\lambda}{R} =
+ \frac{Q'R^2}{R} + \frac{Q(2ax + b)}{2R} + \frac{\lambda}{R} =
+ \frac{Q'R^2 + \frac{1}{2} Q(2ax + b) + \lambda}{R}
+ \end{gather*}
+
+ Надо бы ещё доказать, что такое $Q$ всегда найдется, но нам достаточно того, что в задаче такой $Q$ есть.
+
+ Тогда по лемме нам надо разложить $x^8$ на слагаемые $Q'(x^2 + 1) + Qx + \lambda$
+ для некоторого $Q$.
+
+ Пусть $Q = a_7x^7 + \ldots + a_0, \quad Q' = 7a_7x^6 + \ldots + a_1$
+
+ Получится система линейных уравнений, которую я выписывать не буду, а выпишу сразу ответ:
+ \[
+ Q = \frac{1}{8}x^7 - \frac{7}{48} x^5 + \frac{35}{192} x^3 - \frac{35}{128} x, \quad \lambda = \frac{35}{128}
+ \]
+
+ Тогда
+
+ \begin{align*}
+ \int \frac{x^8 \dif x}{\sqrt{x^2 + 1}} &= \br{
+ \frac{1}{8}x^7 - \frac{7}{48} x^5 + \frac{35}{192} x^3 - \frac{35}{128} x
+ } \sqrt{x^2 + 1} + \frac{25}{128}\int \frac{\dif x}{\sqrt{x^2 + 1}} \\[8pt]&=
+ \br{
+ \frac{1}{8}x^7 - \frac{7}{48} x^5 + \frac{35}{192} x^3 - \frac{35}{128} x
+ } \sqrt{x^2 + 1} + \frac{25}{128} \log \left|x + \sqrt{x^2 + 1}\right| + C
+ \end{align*}
+
+\question{17.b}{
+ \[
+ \int \frac{(x + 3) \dif x}{(x^2 + 1)\sqrt{x^2 + x + 1}}
+ \]
+}
+
+ \newcommand{\brac}[2]{ \br{ \dfrac{#1}{#2} } }
+ \newcommand{\sbrac}[2]{ \br{ \frac{#1}{#2} } }
+ \renewcommand{\deft}{\br{ \sqrt{x^2 + x + 1} - x }}
+
+ \begin{gather*}
+ \sqrt{x^2 + x + 1} = x + t\\
+ x^2 + x + 1 = x^2 + 2xt + t^2
+ \end{gather*}
+ \begin{align*}
+ x &= \frac{t^2 - 1}{1 - 2t}\\
+ \dif x &= -\frac{2(t^2 - t + 1)}{(1 - 2t)^2} \dif t\\
+ x + t &= \frac{t^2 - t + 1}{2t - 1}\\
+ x + 3 &= \frac{t^2 - 6t + 2}{1 - 2t}\\
+ x^2 + 1 &= \frac{t^4 + 2t^2 - 4t + 2}{(1 - 2t)^2}
+ \end{align*}
+
+ \begin{align*}
+ \int \frac{(x + 3) \dif x}{(x^2 + 1)\sqrt{x^2 + x + 1}}
+ &= -2 \int \frac{(x + 3)(t^2 - t + 1)\dif t}{(x^2 + 1)(x + t)(1 - 2t)^2}\\[8pt]
+ &= -2 \int \frac{
+ \brac{t^2 - 6t + 2}{1 - 2t}(t^2 - t + 1) \dif t
+ }{
+ \brac{t^4 + 2t^2 - 4t + 2}{(1 - 2t)^2}
+ \brac{t^2 - t + 1}{2t - 1}
+ (1 - 2t)^2
+ }\\[8pt]
+ &= -2 \int - \frac{t^2 - 6t + 2}{t^4 + 2t^2 - 4t + 2} \dif t\\[8pt]
+ &= 2 \int \frac{t^2 - 6t + 2}{t^4 + 2t^2 - 4t + 2} \dif t\\[8pt]
+ &= \frac{1}{\sqrt{2}} \int \br{
+ \frac{2 t - 3 \sqrt2 + 4}{-t^2 + \sqrt2 t + \sqrt2 - 2}
+ + \frac{2 t + 3 \sqrt2 + 4}{t^2 + \sqrt2 t + \sqrt2 + 2}
+ } \dif t\\[8pt]
+ \end{align*}
+
+ \renewcommand{\deft}{t}
+ \def\firstdenum{\br{ \deft - \frac{\sqrt2}{2} }^2 + \sbrac{\sqrt{6 - 4\sqrt2}}{2}^2}
+ \def\firstpoly{ \deft^2 - \sqrt2 \deft - \sqrt2 + 2 }
+ \def\firstsqrt{ \sqrt{6 - 4\sqrt2} }
+
+ \def\seconddenum{ \br{ \deft + \frac{\sqrt2}{2} }^2 + \sbrac{\sqrt{6 + 4\sqrt2}}{2}^2 }
+ \def\secondpoly{ \deft^2 + \sqrt2 \deft + \sqrt2 + 2 }
+ \def\secondsqrt{ \sqrt{6 + 4\sqrt2} }
+
+ \begin{align*}
+ \int \frac{2 t - 3 \sqrt2 + 4}{-t^2 + \sqrt2 t + \sqrt2 - 2}\dif t
+ &= -2 \int \frac{t - \frac{3\sqrt2}{2} + 2}{\firstdenum}\dif t\\[8pt]
+ &= -2 \int \frac{t - \frac{\sqrt2}{2}}{\firstdenum} \dif t
+ -2 \int \frac{-\sqrt2 + 2}{\firstdenum} \dif t\\[8pt]
+ &= - \int \frac{\dif \br{ \firstpoly }}{\firstpoly} \dif t
+ +2(\sqrt2 - 2) \frac{2}{\firstsqrt} \arctan \frac{2t - \sqrt2}{\firstsqrt}\\[8pt]
+ &= -\log{ \left|\firstpoly\right| } -
+ 4\arctan \frac{2t - \sqrt2}{\firstsqrt} + C_1
+ \end{align*}
+
+ \begin{align*}
+ \int \frac{2 t + 3 \sqrt2 + 4}{\secondpoly}\dif t
+ &= \int \frac{2t + 3\sqrt2 + 4}{\seconddenum} \dif t\\[8pt]
+ &= \int \frac{2t + \sqrt2}{\seconddenum} +
+ \int \frac{2\sqrt2 + 4}{\seconddenum}\\[8pt]
+ &= \log{ \left|\secondpoly\right| } +
+ 2(\sqrt2 + 2)\frac{2}{\secondsqrt} \arctan \frac{2t + \sqrt2}{\secondsqrt} + C_2\\[8pt]
+ &= \log{ \left|\secondpoly\right| } +
+ 4\arctan \frac{2t + \sqrt2}{\secondsqrt} + C_2
+ \end{align*}
+
+ \begin{align*}
+ \int \frac{(x + 3) \dif x}{(x^2 + 1)\sqrt{x^2 + x + 1}}
+ &= -\log{ \left|\firstpoly\right| } -
+ 4\arctan \frac{2t - \sqrt2}{\firstsqrt}\\[8pt]
+ & +\log{ \left|\secondpoly\right| } +
+ 4\arctan \frac{2t + \sqrt2}{\secondsqrt} + C\\[8pt]
+ \end{align*}
+ \renewcommand{\deft}{\br{\sqrt{x^2 + x + 1} - x}}
+ \begin{align*}
+ &= -\log{ \left|\firstpoly\right| } -
+ 4\arctan \frac{2\deft - \sqrt2}{\firstsqrt}\\[8pt]
+ & +\log{ \left|\secondpoly\right| } +
+ 4\arctan \frac{2\deft + \sqrt2}{\secondsqrt} + C
+ \end{align*}
+
+\clearpage
+\question{17.c}{
+ \[
+ \int \frac{\dif x}{(x^3 - x)\sqrt{x^2 + x + 4}}
+ \]
+}
+ \begin{gather*}
+ \sqrt{x^2 + x + 4} = x + t\\
+ x^2 + x + 4 = x^2 + 2xt + t^2
+ \end{gather*}
+ \begin{align*}
+ x &= \frac{t^2 - 4}{1 - 2t}\\
+ \dif x &= -\frac{2(t^2 - t + 4)}{(1 - 2t)^2} \dif t\\
+ x + t &= \frac{t^2 - t + 4}{2t - 1}
+ \end{align*}
+
+ \medskip
+
+ \renewcommand{\deft}{\sqrt{x^2 + x + 4} - x}
+
+ \begin{align*}
+ \int \frac{\dif x}{(x^3 - x)\sqrt{x^2 + x + 4}}
+ &= \int \frac{\dif x}{x(x + 1)(x - 1)\sqrt{x^2 + x + 4}}\\[8pt]
+ &= -\int \frac{2(t^2 - t + 4) \dif t}{(1 - 2t)^2 x(x + 1)(x - 1)(x + t)}\\[8pt]
+ &= -2\int \frac{(t^2 - t + 4) \dif t}{
+ (1 - 2t)^2
+ \brac{t^2 - 4}{1 - 2t}
+ \brac{t^2 - 5 + 2t}{1 - 2t}
+ \brac{t^2 - 3 - 2t}{1 - 2t}
+ \brac{t^2 - t + 4}{2t - 1}
+ }\\[8pt]
+ &= 2 \int \frac{(1 - 2t)^2 \dif t}{
+ (t^2 - 4)(t^2 + 2t - 5)(t^2 - 2t - 3)
+ }\\[8pt]
+ &= 2 \int \frac{(1 - 2t)^2 \dif t}{
+ (t - 2)(t + 2)(t - \sqrt{6} + 1)(t + \sqrt{6} + 1)(t + 1)(t - 3)
+ }\\[8pt]
+ &= 2\int \br{
+ -\frac{1}{t^2 - 4}
+ -\frac{1}{8(t + 1)}
+ +\frac{1}{8(t - 3)}
+ -\frac{1}{4\sqrt{6}(t + \sqrt{6} + 1)}
+ +\frac{1}{4\sqrt{6}(t - \sqrt{6} + 1)}
+ } \dif t\\[8pt]
+ &= -\arctan \frac{t}{2}
+ -\frac{1}{4}\log |t + 1|
+ +\frac{1}{4}\log |t - 3|
+ -\frac{1}{2\sqrt{6}}\log |t + \sqrt{6} + 1|
+ +\frac{1}{2\sqrt{6}}\log |t - \sqrt{6} + 1|\\[8pt]
+ &= -\arctan \frac{\deft}{2}
+ -\frac{1}{4}\log |\deft + 1|
+ +\frac{1}{4}\log |\deft - 3|\\[8pt]
+ &-\frac{1}{2\sqrt{6}}\log |\deft + \sqrt{6} + 1|
+ +\frac{1}{2\sqrt{6}}\log |\deft - \sqrt{6} + 1|\\[8pt]
+ \end{align*}
+
+\end{document}
generated by cgit v1.2.1-18-gbd029 (git 2.20.0) at 2025-07-30 17:19:59 +0000