1. 定积分的概念与性质
1. 定积分的定义
设 \(f\left( x\right)\) 是定义在区间 \(\left\lbrack {a,b}\right\rbrack\) 上的有界函数,任取分点 \(a = {x}_{0} < {x}_{1} < {x}_{2} <\) \(\cdots < {x}_{n} = b\) ,将 \(\left\lbrack {a,b}\right\rbrack\) 分为 \(n\) 个子区间 \(\left\lbrack {{x}_{i - 1},{x}_{i}}\right\rbrack\) ,记 \(\Delta {x}_{i} = {x}_{i} - {x}_{i - 1}\left( {i = 1,2,\cdots ,n}\right)\) ,又在每个子区间上任取一点 \({\xi }_{i} \in \left\lbrack {{x}_{i - 1},{x}_{i}}\right\rbrack \left( {i = 1,2,\cdots ,n}\right)\) ,若不论对区间 \(\left\lbrack {a,b}\right\rbrack\) 如何分法,也不论 \({\xi }_{i}\) 在 \(\left\lbrack {x}_{i - 1}\right.\) , \(\left. {x}_{i}\right\rbrack\) 中如何取法,只要当 \(\lambda = \mathop{\max }\limits_{{1 \leq i \leq n}}\Delta {x}_{i}\) 趋于零时,和式 \(\mathop{\sum }\limits_{{i = 1}}^{n}f\left( {\xi }_{i}\right) \Delta {x}_{i}\) 的极限存在,则称此极限值为 \(f\left( x\right)\) 在 \(\left\lbrack {a,b}\right\rbrack\) 上的定积分,记为\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = \mathop{\lim }\limits_{{\lambda \rightarrow 0}}\mathop{\sum }\limits_{{i = 1}}^{n}f\left( {\xi }_{i}\right) \Delta {x}_{i}\]此时也称 \(f\left( x\right)\) 在 \(\left\lbrack {a,b}\right\rbrack\) 上可积.
特别地,把区间 \(\left\lbrack {a,b}\right\rbrack\) 分为 \(n\) 等份, \({\xi }_{i}\) 取为每个小区间的右端点,则有\[\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{b - a}{n}\mathop{\sum }\limits_{{i = 1}}^{n}f\left( {a + \frac{b - a}{n}i}\right) = {\int }_{a}^{b}f\left( x\right) \mathrm{d}x\]\[\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\mathop{\sum }\limits_{{i = 1}}^{n}f\left( \frac{i}{n}\right) = {\int }_{0}^{1}f\left( x\right) \mathrm{d}x\text{(此时 }a = 0,b = 1\text{ ) }\]使用以上两个公式可计算某些和式的极限.
2. 定积分的基本性质
(1)定积分的结果与积分变量无关,即 \(\displaystyle{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = {\int }_{a}^{b}f\left( t\right) \mathrm{d}t\)
(2)\(\displaystyle{\int }_{a}^{a}f\left( x\right) \mathrm{d}x \equiv 0\)
(3)\(\displaystyle{\int }_{b}^{a}f\left( x\right) \mathrm{d}x = - {\int }_{a}^{b}f\left( x\right) \mathrm{d}x\)
(4)若\(\displaystyle f\left( x\right)\) 在 \(\left\lbrack {a,b}\right\rbrack\) 上可积, \(k\) 为任一常数,则 \(\displaystyle{\int }_{a}^{b}{kf}\left( x\right) \mathrm{d}x = k{\int }_{a}^{b}f\left( x\right) \mathrm{d}x\)
(5)若\(\displaystyle f\left( x\right) \text{ 、 }g\left( x\right)\) 在 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 上都可积,则\[{\int }_{a}^{b}\left\lbrack {f\left( x\right) \pm g\left( x\right) }\right\rbrack \mathrm{d}x = {\int }_{a}^{b}f\left( x\right) \mathrm{d}x \pm {\int }_{a}^{b}g\left( x\right) \mathrm{d}x\](6)设函数 \(\displaystyle f\left( x\right)\) 在 \(\displaystyle\left\lbrack {a,c}\right\rbrack ,\left\lbrack {c,b}\right\rbrack ,\left\lbrack {a,b}\right\rbrack\) 上都可积,则\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = {\int }_{a}^{c}f\left( x\right) \mathrm{d}x + {\int }_{c}^{b}f\left( x\right) \mathrm{d}x\]当 \(c\) 点在 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 外时,结论仍成立
(7)设 \(\displaystyle f\left( x\right) \text{ 、 }g\left( x\right)\) 在 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 上可积,且满足不等式 \(\displaystyle f\left( x\right) \leq g\left( x\right) ,x \in \left\lbrack {a,b}\right\rbrack\) ,则\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x \leq {\int }_{a}^{b}g\left( x\right) \mathrm{d}x\](8)估值定理 设 \(\displaystyle f\left( x\right)\) 在 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 上的最大值、最小值分别为 \(M\) 和 \(m\) ,则有\[m\left( {b - a}\right) \leq {\int }_{a}^{b}f\left( x\right) \mathrm{d}x \leq M\left( {b - a}\right)\](9) 积分中值定理 设 \(\displaystyle f\left( x\right)\) 在 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 上连续,则在 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 上至少存在一点 \(\xi\) ,使得\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = f\left( \xi \right) \left( {b - a}\right)\]称 \(\displaystyle\frac{1}{b - a}{\int }_{a}^{b}f\left( x\right) \mathrm{d}x\) 为函数 \(\displaystyle f\left( x\right)\) 在 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 上的积分平均值.
3. 积分不等式
设\(\displaystyle f\left( x\right) \text{ 、 }g\left( x\right)\) 在区间 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 上可积,则有下列不等式
(1)\(\displaystyle\left| {{\int }_{a}^{b}f\left( x\right) \mathrm{d}x}\right| \leq {\int }_{a}^{b}\left| {f\left( x\right) }\right| \mathrm{d}x\)
(2)许瓦尔兹不等式\[{\left\lbrack {\int }_{a}^{b}f\left( x\right) g\left( x\right) \mathrm{d}x\right\rbrack }^{2} \leq {\int }_{a}^{b}{\left\lbrack f\left( x\right) \right\rbrack }^{2}\mathrm{\;d}x \vdot {\int }_{a}^{b}{\left\lbrack g\left( x\right) \right\rbrack }^{2}\mathrm{\;d}x\]
2. 微积分基本公式
1. 变上限定积分
(1)若\(f\left( x\right)\) 在 \(\left\lbrack {a,b}\right\rbrack\) 上连续,则 \(\displaystyle\Phi \left( x\right) = {\int }_{a}^{x}f\left( t\right) \mathrm{d}t\) 在 \(\displaystyle\left\lbrack {a,b}\right\rbrack\) 上可导,且有\[{\Phi }^{\prime }\left( x\right) = \frac{\mathrm{d}}{\mathrm{d}x}{\int }_{a}^{x}f\left( t\right) \mathrm{d}t = f\left( x\right)\](2)若\(f\left( x\right)\) 在 \(\left\lbrack {a,b}\right\rbrack\) 上连续, \(g\left( x\right)\) 是可微的,\[\frac{\mathrm{d}}{\mathrm{d}x}\left( {{\int }_{a}^{g\left( x\right) }f\left( t\right) \mathrm{d}t}\right) = f\left\lbrack {g\left( x\right) }\right\rbrack {g}^{\prime }\left( x\right)\](3)若上、下限都是 \(x\) 的可微函数,则\[\frac{\mathrm{d}}{\mathrm{d}x}\left( {{\int }_{a\left( x\right) }^{b\left( x\right) }f\left( t\right) \mathrm{d}t}\right) = f\left\lbrack {b\left( x\right) }\right\rbrack {b}^{\prime }\left( x\right) - f\left\lbrack {a\left( x\right) }\right\rbrack {a}^{\prime }\left( x\right)\]实际上, 这是一个求复合函数的导数问题.
2. 定积分和不定积分的关系
(1)原函数存在定理 若函数 \(f\left( x\right)\) 在 \(\left\lbrack {a,b}\right\rbrack\) 上连续,则函数 \(\displaystyle\Phi \left( x\right) = {\int }_{a}^{x}f\left( t\right) \mathrm{d}t\) 是 \(f\left( x\right)\) 在 \(\lbrack a\) , \(b\rbrack\) 区间上的一个原函数.
(2)牛顿一莱布尼兹公式 若 \(\displaystyle F\left( x\right)\) 是 \(\displaystyle f\left( x\right)\) 在区间 \(\left\lbrack {a,b}\right\rbrack\) 上的一个原函数,而且 \(f\left( x\right)\) 在 \(\lbrack a\) , \(b\rbrack\) 上连续,则\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = F\left( b\right) - F\left( a\right)\]这个公式也称为微积分基本公式, 它指出了定积分与不定积分的内在联系.
3. 定积分的换元法和分部积分法
1. 换元积分法
若函数 \(f\left( x\right)\) 在区间 \(\left\lbrack {a,b}\right\rbrack\) 上连续; 函数 \(x = \varphi \left( t\right)\) 在区间 \(\left\lbrack {\alpha ,\beta }\right\rbrack\) 上单调且具有连续导数,当 \(\alpha \leq t \leq \beta\) 时, \(a \leq \varphi \left( t\right) \leq b\) ,且 \(\varphi \left( \alpha \right) = a,\varphi \left( \beta \right) = b\) ,则有定积分的换元公式\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = {\int }_{\alpha }^{\beta }f\left\lbrack {\varphi \left( t\right) }\right\rbrack {\varphi }^{\prime }\left( t\right) \mathrm{d}t\]2. 分部积分法
设函数 \(u\left( x\right) ,v\left( x\right)\) 在区间 \(\left\lbrack {a,b}\right\rbrack\) 上具有连续导数 \(\displaystyle {u}^{\prime }\left( x\right) \text{ 、 }{v}^{\prime }\left( x\right)\) ,则有定积分的分部积分公式\[{\int }_{a}^{b}u\left( x\right) {v}^{\prime }\left( x\right) \mathrm{d}x = {\left. \left\lbrack u\left( x\right) v\left( x\right) \right\rbrack \right| }_{a}^{b} - {\int }_{a}^{b}v\left( x\right) {u}^{\prime }\left( x\right) \mathrm{d}x\]3. 常用公式
设\(f\left( x\right)\) 为连续函数
(1)\(\displaystyle{\int }_{-a}^{a}f\left( x\right) \mathrm{d}x = {\int }_{0}^{a}\left\lbrack {f\left( x\right) + f\left( {-x}\right) }\right\rbrack \mathrm{d}x\)
(2)\(\displaystyle{\int }_{-a}^{a}f\left( x\right) \mathrm{d}x = \left\{ \begin{array}{ll} 2{\displaystyle\int }_{0}^{a}f\left( x\right) \mathrm{d}x, & f\left( x\right) \text{ 是偶函数 } \\ 0, & f\left( x\right) \text{ 是奇函数 } \end{array}\right.\)
(3)\(\displaystyle{\int }_{0}^{\frac{\pi }{2}}f\left( {\sin x}\right) \mathrm{d}x = {\int }_{0}^{\frac{\pi }{2}}f\left( {\cos x}\right) \mathrm{d}x\)
(4)\(\displaystyle{\int }_{0}^{\pi }{xf}\left( {\sin x}\right) \mathrm{d}x = \frac{\pi }{2}{\int }_{0}^{\pi }f\left( {\sin x}\right) \mathrm{d}x\)
(5)\(\displaystyle f\left( {x + L}\right) = f\left( x\right) ,\left( {L > 0}\right)\) ,则 \(\displaystyle{\int }_{0}^{L}f\left( x\right) \mathrm{d}x = {\int }_{-\frac{L}{2}}^{\frac{L}{2}}f\left( x\right) \mathrm{d}x = {\int }_{a}^{a + L}f\left( x\right) \mathrm{d}x\)
(6)\(\displaystyle{\int }_{0}^{\frac{\pi }{2}}{\left( \sin x\right) }^{n}\mathrm{\;d}x = {\int }_{0}^{\frac{\pi }{2}}{\left( \cos x\right) }^{n}\mathrm{\;d}x = \left\{ \begin{array}{ll} \displaystyle\frac{\left( {n - 1}\right) !!}{n!!} \vdot \frac{\pi }{2}, & \text{ 当 }n\text{ 为偶数时 } \\ \displaystyle\frac{\left( {n - 1}\right) !!}{n!!}, & \text{ 当 }n\text{ 为奇数时 } \end{array}\right.\)
此公式在定积分计算中十分有用,应记住. 当 \(n\) 为偶数时, \(n!\) ! 表示所有偶数 (不大于 \(n\) ) 连乘积. \(n\) 为奇数时, \(n!\) 表示所有奇数 (不大于 \(n\) ) 的连乘积.
4. 广义积分
1. 无穷区间上的广义积分
设函数 \(f\left( x\right)\) 在区间 \(\lbrack a, + \infty )\) 上有定义,在 \(\left\lbrack {a,b}\right\rbrack \left( {b < + \infty }\right)\) 上可积,若极限 \(\displaystyle\mathop{\lim }\limits_{{b \rightarrow + \infty }}{\int }_{a}^{b}f\left( x\right) \mathrm{d}x\) 存在,则定义\[{\int }_{a}^{+\infty }f\left( x\right) \mathrm{d}x = \mathop{\lim }\limits_{{b \rightarrow + \infty }}{\int }_{a}^{b}f\left( x\right) \mathrm{d}x\]并称 \(\displaystyle{\int }_{a}^{+\infty }f\left( x\right) \mathrm{d}x\) 为 \(f\left( x\right)\) 在 \(\lbrack a, + \infty )\) 上的广义积分,这时也称广义积分 \(\displaystyle{\int }_{a}^{+\infty }f\left( x\right) \mathrm{d}x\) 存在或收敛; 若上述极限不存在,则称广义积分 \(\displaystyle{\int }_{a}^{+\infty }f\left( x\right) \mathrm{d}x\) 不存在或发散.
类似地, 定义\[{\int }_{-\infty }^{b}f\left( x\right) \mathrm{d}x = \mathop{\lim }\limits_{{a \rightarrow - \infty }}{\int }_{a}^{b}f\left( x\right) \mathrm{d}x\]\[{\int }_{-\infty }^{+\infty }f\left( x\right) \mathrm{d}x = {\int }_{-\infty }^{c}f\left( x\right) \mathrm{d}x + {\int }_{c}^{+\infty }f\left( x\right) \mathrm{d}x = \mathop{\lim }\limits_{{a \rightarrow - \infty }}{\int }_{a}^{c}f\left( x\right) \mathrm{d}x + \mathop{\lim }\limits_{{b \rightarrow + \infty }}{\int }_{c}^{b}f\left( x\right) \mathrm{d}x\]2. 无界函数的广义积分 (瑕积分)
设函数 \(f\left( x\right)\) 在 \(\lbrack a,b)\) 上连续,而且 \(\mathop{\lim }\limits_{{x \rightarrow {b}^{ - }}}f\left( x\right) = \infty\) ,若极限 \(\displaystyle\mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ + }}}{\int }_{a}^{b - \varepsilon }f\left( x\right) \mathrm{d}x\) 存在,则定义\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ + }}}{\int }_{a}^{b - \varepsilon }f\left( x\right) \mathrm{d}x\]并称 \(\displaystyle{\int }_{a}^{b}f\left( x\right) \mathrm{d}x\) 为 \(f\left( x\right)\) 在 \(\lbrack a,b)\) 上的广义积分,这时也称广义积分 \(\displaystyle{\int }_{a}^{b}f\left( x\right) \mathrm{d}x\) 存在或收敛; 若上述极限不存在,则称广义积分 \(\displaystyle{\int }_{a}^{b}f\left( x\right) \mathrm{d}x\) 不存在或发散.
类似地,若 \(f\left( x\right)\) 在 \((a,b\rbrack\) 上连续, \(\mathop{\lim }\limits_{{x \rightarrow {a}^{ + }}}f\left( x\right) = \infty\) ,则定义\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ + }}}{\int }_{a + \varepsilon }^{b}f\left( x\right) \mathrm{d}x\]若\(f\left( x\right)\) 在(a, b)内连续, \(\mathop{\lim }\limits_{{x \rightarrow {a}^{ + }}}f\left( x\right) = \infty ,\mathop{\lim }\limits_{{x \rightarrow {b}^{ - }}}f\left( x\right) = \infty\) ,则定义\[{\int }_{a}^{b}f\left( x\right) \mathrm{d}x = \mathop{\lim }\limits_{{{\varepsilon }_{1} \rightarrow {0}^{ + }}}{\int }_{a + {\varepsilon }_{1}}^{c}f\left( x\right) \mathrm{d}x + \mathop{\lim }\limits_{{{\varepsilon }_{2} \rightarrow {0}^{ + }}}{\int }_{c}^{b - {\varepsilon }_{2}}f\left( x\right) \mathrm{d}x\]
5. 反常积分审敛
使用\(\displaystyle\int\frac1{x^p}\mathrm{d}x,\int\frac{\ln x}{x^p}\mathrm{d}x\)的基本结论:
(1)\(\displaystyle\int_0^1\frac1{x^p}\mathrm{d}x\begin{cases}\text{收敛,}0<p<1,\\\text{发散,}p\geqslant1;&\end{cases}\)
(2)\(\displaystyle\int_0^1\frac{\ln x}{x^p} \mathrm{d}x\begin{cases}\text{收敛,}0<p<1,\\\text{发散,}p\geqslant1;\end{cases}\)
(3)\(\displaystyle\int_{1}^{+\infty}\frac{1}{x^p} \mathrm{dx}\begin{cases}\text{收敛,}p>1,\\\text{发散,}p\leqslant1;\end{cases}\)
(4)\(\displaystyle\int_{1}^{+\infty}\frac{\ln x}{x^p} \mathrm{dx}\begin{cases}\text{收敛,}p>1,\\\text{发散,}p\leqslant1\end{cases}\)
二级结论:
①\(\displaystyle\int_1^2\frac1{x\ln^px} \mathrm{d}x\begin{cases}\text{收敛,}0<p<1,\\\text{发散,}p\geqslant1;\end{cases}\)
②\(\displaystyle\int_{2}^{+\infty}\frac{1}{x\ln^{p}x}\mathrm{d}x\begin{cases}\text{收敛,}p>1,\\\text{发散,}p\leqslant1;\end{cases}\)
③\(\displaystyle\int_{1}^{+\infty}\frac1{x\ln^{p}x}\mathrm{d}x\text{必发散}\)
④\(\displaystyle\int_A^{+\infty}\mathrm{e}^{-ax}\vdot f(x)\mathrm{d}x\begin{cases}\text{收敛,}a>0,\\\text{发散,}a<0\end{cases}\)
⑤\(\displaystyle\int_1^2\frac1{x^p\ln^qx}\mathrm{d}x\begin{cases}\text{收敛,}0<q<1,\\\text{发散,}q\geqslant1;\end{cases}\)
⑥\(\displaystyle\int_{2}^{+\infty}\frac{1}{x^{p}\ln^{q}x}\mathrm{d}x\begin{cases}\text{收敛,}p>1,&q\text{任意,}\\\text{发散,}p<1,&q\text{任意,}\\\text{收敛,}p=1,&q>1,\\\text{发散,}p=1,&q\leqslant1\end{cases}\)
⑦\(\displaystyle\int_0^1\frac{1}{\left|\ln x\right|^p} \mathrm{d}x\begin{cases}\text{收敛,}0<p<1,\\\text{发散,}p\geqslant1;\end{cases}\)
⑧\(\displaystyle\int_1^2\frac{1}{\left|\ln x\right|^p} \mathrm{d}x\begin{cases}\text{收敛,}0<p<1,\\\text{发散,}p\geqslant1\end{cases}\)
⑨\(\displaystyle\int_{0}^{2}\frac{1}{\left|\ln x\right|^{p}}\mathrm{d}x\begin{cases}\text{收敛,}0<p<1,\\\text{发散,}p\geqslant1;\end{cases}\)
⑩\(\displaystyle\int_{2}^{+\infty}\frac{1}{\left|\ln x\right|^{p}}\mathrm{d}x\text{必发散}\)