定积分的算法有两种:
换元积分法
如果
;x=ψ(t)在[α,β]上单值、可导;当α≤t≤β时,a≤ψ(t)≤b,且ψ(α)=a,ψ(β)=b,
则
![](https://iknow-pic.cdn.bcebos.com/03087bf40ad162d9df06bc4d1cdfa9ec8b13cdf3?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
分部积分法
设u=u(x),v=v(x)均在区间[a,b]上可导,且u′,v′∈R([a,b]),则有分部积分公式:
![](https://iknow-pic.cdn.bcebos.com/79f0f736afc379310bc25878e6c4b74543a9113c?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
![](https://iknow-pic.cdn.bcebos.com/11385343fbf2b211f8515c1fc78065380cd78e60?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
扩展资料
定积分的性质:
1、当a=b时,![](https://iknow-pic.cdn.bcebos.com/79f0f736afc379310afe5f78e6c4b74542a911c8?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
2、当a>b时, ![](https://iknow-pic.cdn.bcebos.com/b21bb051f819861825d136d547ed2e738ad4e6c9?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
3、常数可以提到积分号前。![](https://iknow-pic.cdn.bcebos.com/dcc451da81cb39db5768be07dd160924ab183033?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
4、代数和的积分等于积分的代数和。![](https://iknow-pic.cdn.bcebos.com/aa18972bd40735fa0f0c97e893510fb30e2408e7?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
5、定积分的可加性:如果积分区间[a,b]被c分为两个子区间[a,c]与[c,b]则有![](https://iknow-pic.cdn.bcebos.com/ac6eddc451da81cbd9a85a9c5f66d01609243111?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
又由于性质2,若f(x)在区间D上可积,区间D中任意c(可以不在区间[a,b]上)满足条件。
6、如果在区间[a,b]上,f(x)≥0,则![](https://iknow-pic.cdn.bcebos.com/fd039245d688d43f51e748a3701ed21b0ef43b11?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
7、积分中值定理:设f(x)在[a,b]上连续,则至少存在一点ε在(a,b)内使![](https://iknow-pic.cdn.bcebos.com/7a899e510fb30f24ac663552c595d143ac4b03e7?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)