不得不吐槽一下有时做题看答案时,答案总是列个公式就给结果了(公式我也会列啊,但我就是不会算啊🙄

比如今天做到的:

02πx2+1dx\int_0^{2\pi} {\sqrt{x^2+1}} \,{\rm d}x

通过换元(令x=tan(t)x=tan(t))后可以转化为求:

sec3(t)dt\int {sec^3(t)} \,{\rm d}t

然后就不会搞了,查Wlofram时,Wolfram又给了一个公式(同样是没过程… (╯‵□′)╯︵┻━┻):

secm(u)du=sin(u)secm1(u)m1+m2m1secm2(u)du\int {sec^m(u)} \,{\rm d}u = \frac{sin(u)sec^{m-1}(u)}{m-1} + \frac{m-2}{m-1} \int {sec^{m-2}(u)} \,{\rm d}u

m=3m=3代一下就出结果了,但是问题好像是变复杂了,因为他没给推导过程(也可能只是因为wtcl - -)。所以这里就水个贴写一下这个过程。

首先看Wolfram给的公式的结构大概看出了是用分布积分法的,然后推的时候其实还用到了积分再现,和一个神奇的公式:

tan2x=sec2x1tan^2x = sec^2x-1

首先从比较简单的sec3(x)sec^3(x)开始说,用分部积分的话首先要拆成两部分,因为是3次方,可能性比较大的两种可能就是sec(x)sec(x)sec2(x)sec^2(x)了。然后sec(x)sec(x)用来求导,sec2(x)sec^2(x)用来积分(用的是张宇的表格法,但是LaTeX\LaTeX画表格比较麻烦,而我又比较懒

首先:

(sec(x))=sec(x)tan(x)sec2(x)dx=tan(x)sec(x)dx=lnsec(x)+tan(x)\begin{align} (sec(x))' &= sec(x)tan(x) \\ \int {sec^2(x)} \,{\rm d}x &= tan(x) \\ \int {sec(x)} \,{\rm d}x &= ln|sec(x)+tan(x)| \end{align}

于是:

sec3(t)dt=sec(t)tan(t)sec(t)tan2(t)dt=sec(t)tan(t)sec(t)(sec2(t)1)dt=sec(t)tan(t)(sec3(t)sec(t))dt=sec(t)tan(t)(sec3(t))dt+(sec(t))dt=sec(t)tan(t)+lnsec(x)+tan(x)(sec3(t))dt\begin{align} \int {sec^3(t)} \,{\rm d}t &= sec(t)tan(t) - \int {sec(t)*tan^2(t)} \,{\rm d}t \\ &= sec(t)tan(t) - \int {sec(t)*(sec^2(t)-1)} \,{\rm d}t \\ &= sec(t)tan(t) - \int {(sec^3(t)-sec(t))} \,{\rm d}t \\ &= sec(t)tan(t) - \int {(sec^3(t))} \,{\rm d}t+\int {(sec(t)*)} \,{\rm d}t \\ &= sec(t)tan(t) + ln|sec(x)+tan(x)| - \int {(sec^3(t))} \,{\rm d}t \\ \end{align}

中间用到了哪个神奇的公式,然后积分再现也出来了,把(sec3(t))dt- \int {(sec^3(t))} \,{\rm d}t移到左边,就有:

2sec3(t)dt=sec(t)tan(t)+lnsec(x)+tan(x)sec3(t)dt=sec(t)tan(t)+lnsec(x)+tan(x)2+C\begin{align} 2\int {sec^3(t)} \,{\rm d}t &= sec(t)tan(t) + ln|sec(x)+tan(x)| \\ \int {sec^3(t)} \,{\rm d}t &= \frac{sec(t)tan(t) + ln|sec(x)+tan(x)| }{2} + C \end{align}

然后求secm(x)sec^m(x)也是用类似的方法,拆成secm2(x)sec^{m-2}(x)sec2(x)sec^2(x)secm2(x)sec^{m-2}(x)做求导,sec2(x)sec^2(x)做积分

(sec(x))=(m2)secm3(x)sec(x)tan(x)=(m2)secm2(x)tan(x)sec2(x)dx=tan(x)\begin{align} &(sec(x))' = (m-2)sec^{m-3}(x)*sec(x)tan(x) = (m-2)sec^{m-2}(x)tan(x) \\ &\int {sec^2(x)} \,{\rm d}x = tan(x) \end{align}

分部积分:

secm(t)dt=secm2tan(x)(m2)secm2(t)tan2(t)dt=secm2tan(x)(m2)secm2(t)(sec2(t)1)dt=secm2tan(x)(m2)secm(t)secm2(t)dt=secm2tan(x)+(m2)secm2(t)dt(m2)secm(t)dt\begin{align} \int {sec^m(t)} \,{\rm d}t &= sec^{m-2}tan(x) - (m-2)\int {sec^{m-2}(t)tan^2(t)} \,{\rm d}t \\ &= sec^{m-2}tan(x) - (m-2)\int {sec^{m-2}(t)(sec^2(t)-1)} \,{\rm d}t \\ &= sec^{m-2}tan(x) - (m-2)\int {sec^m(t)-sec^{m-2}(t)} \,{\rm d}t \\ &= sec^{m-2}tan(x) + (m-2)\int {sec^{m-2}(t)} \,{\rm d}t - (m-2)\int {sec^m(t)} \,{\rm d}t \end{align}

积分再现:

secm(t)dt=secm2tan(x)+(m2)secm2(t)dt(m2)secm(t)dt(m1)secm(t)dt=secm2tan(x)+(m2)secm2(t)dtsecm(t)dt=secm2tan(x)+(m2)secm2(t)dtm1+C\begin{align} \int {sec^m(t)} \,{\rm d}t &= sec^{m-2}tan(x) + (m-2)\int {sec^{m-2}(t)} \,{\rm d}t - (m-2)\int {sec^m(t)} \,{\rm d}t \\ (m-1)\int {sec^m(t)} \,{\rm d}t &= sec^{m-2}tan(x) + (m-2)\int {sec^{m-2}(t)} \,{\rm d}t \\ \int {sec^m(t)} \,{\rm d}t &= \frac{sec^{m-2}tan(x) + (m-2)\int {sec^{m-2}(t)} \,{\rm d}t}{m-1} +C \end{align}

( 原文地址:https://0xffff.one/d/642