«求积分 $\int\sec^3x\:\rm{d}x$(带原积分的分部积分)»
by pluvet on Dec 14, 2019

$$ \begin{align} I&=\int\sec^3x\:\rm{d}x\\ &=\int{\color{red}{\sec x}}{\color{blue}\sec^2x}\:\rm{d}x\\ 分部积分法\\ u\ =\ \ \ \ \ \ \ \ \ {\color{red}{\sec x}}, v' = {\color{blue}\sec^2x}\\ u' = \sec x \tan x, v\ = \tan\ x\\ &=uv-\int u'v\rm{d}x\\ & = \sec x \tan x - \int \sec x{\color{red}\tan^2x}\rm{d}x\\ 三角平方关系\\ {\color{red}\tan^2x} = {\color{blue}\sec^2x -1}\\ &= \sec x \tan x - \int \sec x({\color{blue}\sec^2x -1})\rm{d}x\\ 括号展开\\ &= \sec x \tan x - \int \sec^3x-\sec x\rm{d}x\\ (I)&= \sec x \tan x - I + {\color{red}\int\sec x\rm{d}x}\\ I 移项\\ I + I = 2I&= \sec x \tan x + {\color{red}\ln|\sec x+\tan x|}\\ 2 除到右边\\ I&= \frac{1}{2}\left(\sec x \tan x + \ln|\sec x+\tan x|\right)\\ \end{align} $$

菜鸡不装逼, 做题不跳步

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