考研数学常用积分之：含有 $a x$ $+$ $b$ 的积分

一、前言

在本文中，「荒原之梦考研数学」将为同学们总结整理被积函数中含有 “$ax$ $+$ $b$” 以及相关变形形式的积分，这些不是基础的积分公式，也不是一般的习题，但可以作为同学们对积分解题方法的积累。

二、正文

§2.1 没有根号

$$
\begin{aligned}
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int \frac{1}{a x+b} \mathrm{~d} x } \\ \\
= & \frac{1}{a} \ln |a \textcolor{magenta}{x}+b| + \textcolor{pink}{C} \\ \\ \\
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int \frac{x}{a x+b} \mathrm{~d} x } \\ \\
= & \frac{1}{a}(a \textcolor{magenta}{x} + b \textcolor{orangered}{-} b \ln |\textcolor{pink}{c} \textcolor{magenta}{x} + b|) + \textcolor{pink}{C} \\ \\ \\
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int \frac{x}{(a x+b)^{2}} \mathrm{~d} x } \\ \\
= & \frac{1}{a^{2}}\left(\ln |a \textcolor{magenta}{x} + b|+\frac{b}{a \textcolor{magenta}{x} + b}\right) + \textcolor{pink}{C} \\ \\ \\
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int \frac{x^{2}}{a x+b} \mathrm{~d} x } \\ \\
= & \frac{1}{a^{3}}\left[\frac{1}{2}(a \textcolor{magenta}{x}+b)^{2} \textcolor{orangered}{-} 2 b(a \textcolor{magenta}{x} + b)+b^{2} \ln |a \textcolor{magenta}{x} + b|\right] + \textcolor{pink}{C} \\ \\ \\
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int \frac{x^{2}}{(a x+b)^{2}} \mathrm{~d} x } \\ \\
= & \frac{1}{a^{3}}\left(a \textcolor{magenta}{x} + b \textcolor{orangered}{-} 2 b \ln |a \textcolor{magenta}{x} + b| \textcolor{orangered}{-} \frac{b^{2}}{a \textcolor{magenta}{x}+b}\right) + \textcolor{pink}{C} \\ \\ \\
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int \frac{1}{x(a x+b)} \mathrm{~d} x } \\ \\
= & \frac{\textcolor{orangered}{ – } 1}{b} \ln \left|\frac{a \textcolor{magenta}{x} + b}{\textcolor{magenta}{x}}\right| + \textcolor{pink}{C} \\ \\ \\
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int \frac{1}{x^{2}(a x+b)} \mathrm{~d} x } \\ \\
= & \frac{\textcolor{orangered}{-} 1}{b \textcolor{magenta}{x} }+\frac{a}{b^{2}} \ln \left|\frac{a \textcolor{magenta}{x} + b}{\textcolor{magenta}{x}}\right| + \textcolor{pink}{C} \\ \\ \\
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int \frac{1}{x(a x+b)^{2}} \mathrm{~d} x } \\ \\
= & \frac{1}{b(a \textcolor{magenta}{x} + b)}\textcolor{orangered} {-} \frac{1}{b^{2}} \ln \left|\frac{a \textcolor{magenta}{x}+b}{\textcolor{magenta}{x}}\right| + \textcolor{pink}{C} \\ \\ \\
\textcolor{springgreen}{\Large{\boldsymbol{\star}}} & \textcolor{yellow}{ \int(a x+b)^{k} \mathrm{~d} x } \\ \\
= & \frac{1}{a(k + 1)}(a \textcolor{magenta}{x}+b)^{k + 1} + \textcolor{pink}{C} \quad (k \neq \textcolor{orangered}{-} 1)
\end{aligned}
$$

§2.2 有根号

$$
\begin{aligned}
\textcolor{yellow}{\Large{\boldsymbol{\star}}} & \textcolor{springgreen}{ \int \sqrt{ a x + b } \mathrm{~d} x } \\ \\
= & \frac{2}{3a} \sqrt{(a\textcolor{magenta}{x}+b)^{3}} + \textcolor{pink}{C} \\ \\ \\
\textcolor{yellow}{\Large{\boldsymbol{\star}}} & \textcolor{springgreen}{\int x \sqrt{ ax + b } \mathrm{~d} x } \\ \\
= & \frac{2}{ 15a^{2} } ( 3a \textcolor{magenta}{x} \textcolor{orangered}{-} 2b ) \sqrt{ (a \textcolor{magenta}{x} + b )^{3} } + \textcolor{pink}{C} \\ \\ \\
\textcolor{yellow}{\Large{\boldsymbol{\lozenge}}} & \textcolor{springgreen}{ \int x^{2} \sqrt{ ax + b } \mathrm{~d} x } \\ \\
= & \frac{2}{ 105 a^{3} } \left( 15a^{2} \textcolor{magenta}{x^{2}} \textcolor{orangered}{-} 12ab \textcolor{magenta}{x} + 8b^{2} \right) \sqrt{ ( a \textcolor{magenta}{x} + b )^{3} } + \textcolor{pink}{C} \\ \\ \\
\textcolor{yellow}{\Large{\boldsymbol{\star}}} & \textcolor{springgreen}{ \int \frac{x}{ \sqrt{ ax + b } } \mathrm{ ~d } x } \\ \\
= & \frac{2}{ 3a^{2} } ( a\textcolor{magenta}{x} – 2 b ) \sqrt{ a\textcolor{magenta}{x} + b } + \textcolor{pink}{C} \\ \\ \\
\textcolor{yellow}{\Large{\boldsymbol{\lozenge}}} & \textcolor{springgreen}{ \int \frac { x^{2} } { \sqrt{ ax + b } } \mathrm{ ~ d } x } \\ \\
= & \frac{2}{ 15a^{3} } \left( 3a^{2} \textcolor{magenta}{ x^{2} } – 4ab \textcolor{magenta}{x} + 8 b^{2} \right) \sqrt{ a \textcolor{magenta}{x} + b } + \textcolor{pink}{C} \\ \\ \\
\textcolor{yellow}{\Large{\boldsymbol{\star}}} & \textcolor{springgreen}{\int \frac{1}{x\sqrt{a⁢⁢⁢x + b}} \mathrm{~d} x } \\ \\
= & \textcolor{tan}{ \begin{cases}
\frac{1}{\sqrt{b}}\ln \left|\frac{\sqrt{a \textcolor{magenta}{⁢⁢⁢x} + b} \textcolor{orangered}{−} \sqrt{b}}{\sqrt{a⁢⁢ \textcolor{magenta}{⁢x} + b} + \sqrt{b}}\right| + \textcolor{pink}{C}, & b > 0 \\ \\
\frac{2}{\sqrt{\textcolor{orangered}{−} b}}\arctan \sqrt{\frac{a⁢⁢⁢ \textcolor{magenta}{x} + b}{\textcolor{orangered}{−} b}} + \textcolor{pink}{C}⁢⁢⁢, & ⁢b < 0 \end{cases} } \\ \\ \\ \textcolor{yellow}{\Large{\boldsymbol{\lozenge}}} & \textcolor{springgreen}{ \int \frac{1}{ x^{2} \sqrt{ ax+b } } \mathrm{~d} x } \\ \\ = & \textcolor{orangered}{-} \sqrt{ \frac { a \textcolor{magenta}{x} + b }{ b \textcolor{magenta}{x} } } \textcolor{orangered}{-} \frac{a}{2b} \textcolor{tan}{ \int \frac{1}{ \textcolor{magenta}{x} \sqrt{ a \textcolor{magenta}{x} + b } } \mathrm{~d} x } \\ \\ = & \textcolor{orangered}{-} \sqrt{ \frac { a \textcolor{magenta}{x} + b }{ b \textcolor{magenta}{x} } } \textcolor{orangered}{-} \frac{a}{2b} \textcolor{tan}{ \begin{cases} \frac{1}{\sqrt{b}}\ln \left|\frac{\sqrt{a \textcolor{magenta}{⁢⁢⁢x} + b} \textcolor{orangered}{−} \sqrt{b}}{\sqrt{a⁢⁢ \textcolor{magenta}{⁢x} + b} + \sqrt{b}}\right| + \textcolor{pink}{C}, & b > 0 \\ \\
\frac{2}{\sqrt{\textcolor{orangered}{−} b}}\arctan \sqrt{\frac{a⁢⁢⁢ \textcolor{magenta}{x} + b}{\textcolor{orangered}{−} b}} + \textcolor{pink}{C}⁢⁢⁢, & ⁢b < 0 \end{cases} } \\ \\ \\ \textcolor{yellow}{\Large{\boldsymbol{\lozenge}}} & \textcolor{springgreen}{ \int \sqrt{ \frac{ a x + b }{x} } \mathrm{~d} x } \\ \\ = & 2 \sqrt{ a \textcolor{magenta}{x} + b } + b \textcolor{tan}{ \int \frac{1}{ \textcolor{magenta}{x} \sqrt{ a\textcolor{magenta}{x} + b } } \mathrm{~d} x } \\ \\ = & 2 \sqrt{ a \textcolor{magenta}{x} + b } + b \textcolor{tan}{ \begin{cases} \frac{1}{\sqrt{b}}\ln \left|\frac{\sqrt{a \textcolor{magenta}{⁢⁢⁢x} + b} \textcolor{orangered}{−} \sqrt{b}}{\sqrt{a⁢⁢ \textcolor{magenta}{⁢x} + b} + \sqrt{b}}\right| + \textcolor{pink}{C}, & b > 0 \\ \\
\frac{2}{\sqrt{\textcolor{orangered}{−} b}}\arctan \sqrt{\frac{a⁢⁢⁢ \textcolor{magenta}{x} + b}{\textcolor{orangered}{−} b}} + \textcolor{pink}{C}⁢⁢⁢, & ⁢b < 0 \end{cases} } \\ \\ \\ \textcolor{yellow}{\Large{\boldsymbol{\lozenge}}} & \textcolor{springgreen}{ \int \frac{ \sqrt{ ax + b } }{ x^{2} } \mathrm{~d} x } \\ \\ = & \frac{ \textcolor{orangered}{-} \sqrt{a \textcolor{magenta}{x} + b } }{ \textcolor{magenta}{x} } + \frac{a}{2} \textcolor{tan}{ \int \frac{1}{ \textcolor{magenta}{x} \sqrt{ a \textcolor{magenta}{x} + b } } \mathrm{~d} x } \\ \\ = & \frac{ \textcolor{orangered}{-} \sqrt{a \textcolor{magenta}{x} + b } }{ \textcolor{magenta}{x} } + \frac{a}{2} \textcolor{tan}{ \begin{cases} \frac{1}{\sqrt{b}}\ln \left|\frac{\sqrt{a \textcolor{magenta}{⁢⁢⁢x} + b} \textcolor{orangered}{−} \sqrt{b}}{\sqrt{a⁢⁢ \textcolor{magenta}{⁢x} + b} + \sqrt{b}}\right| + \textcolor{pink}{C}, & b > 0 \\ \\
\frac{2}{\sqrt{\textcolor{orangered}{−} b}}\arctan \sqrt{\frac{a⁢⁢⁢ \textcolor{magenta}{x} + b}{\textcolor{orangered}{−} b}} + \textcolor{pink}{C}⁢⁢⁢, & ⁢b < 0
\end{cases} }
\end{aligned}
$$

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