不定积分王者100题 |
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1.前言 这不是练习题！是适合那些不定积分熟手的段位题。 如果你会做其中正文的3%，你就足以应付相关考试。10%——不定积分后浪；60%——不定积分高手；85%——MMA/WA;能完成100%的，可以称之为——王者！因为是(从很多地方)收集的一些不定积分题，故大多（划去）只给题目。 其中标记为灰色的题表示可能会进行修改和补充。标记为蓝色的表示作者已经写过答案/已进行讨论。欢迎加入q群：1046154546（1群）666081508（2群） 交流讨论不定积分。目前王者百题尚未完善，欢迎大佬补充更多好的不定积分题。版本号：V2.22.01.11如果你想验证/查看答案，可以利用下面的工具网站。例：输入‘int x’就是对 f(x)=x 求不定积分。但对于特别刁钻的题是无法获得答案的。 一杯茶，一包烟，一道积分做一天！2.热身时间开餐前的一些点心~\int{\frac{1+x^2}{1+x^4}}\text{d}x \int{\frac{\text{d}x}{\left( 1+x \right) \left( 1+x^2 \right)}} \int{\frac{\text{d}x}{\left( 1+x^3 \right) \left( 1+x^2 \right)}} \color{blue}{\int{\frac{\text{d}x}{\sqrt[3]{1-x^3}}} } \int{\frac{\text{d}x}{\lambda +\sqrt{1-x^2}}} \int{\frac{\text{d}x}{a\sin x+b\cos x}} \color{blue}{ \int{\ln \left( \sqrt{1+x}+\sqrt{1-x} \right) \text{dx}} } \int{\ln \left( \sqrt{1-x}-\sqrt{x} \right) \text{d}x} \int{\frac{\sin x+\cos x}{1-\sin x\cos x}\text{d}x} \int{\frac{1}{\sqrt{1+e^x}+\sqrt{1-e^x}}\text{d}x} \int{\frac{\sqrt{1+e^{2x}}}{1+e^x}\mathrm{d}x} \color{blue}{\int{\frac{x+\sqrt{1-x^2}}{1-x\sqrt{1-x^2}}\text{d}x} } \color{blue}{\int{\frac{\text{d}x}{\sec x+\csc x+\cot x+\tan x}}} \color{blue}{\int{x^4\sqrt{\frac{1-x}{1+x}}\text{d}x} } \int{\sqrt{x+\sqrt{x}}\text{d}x} \int{\frac{\sqrt[4]{1+\sqrt[3]{x}}}{x}\mathrm{d}x} \int{\frac{\text{d}x}{x^n\left( 1+x^2 \right)}} \int{\frac{1-\ln x}{\left( x-\ln x \right) ^2}\text{d}x} \int{\frac{x+1+\ln x}{\left( x+1 \right) ^2+\left( x\ln x \right) ^2}\text{d}x} \int{\mathrm{arc}\tan \left( x^{\frac{2}{3}} \right) \mathrm{d}x} \int{\tan x\tan \left( a+x \right) \text{d}x} \int{\cot \left( x-a \right) \cot \left( x-b \right) \mathrm{d}x} \int{e^{\frac{x}{2}}\frac{\cos x}{\sqrt{\cos x+\sin x}}\text{d}x} \int{\frac{\text{dx}}{\sqrt{\tan ^2x+2}}} \color{blue}{\int{\sqrt{\tan ^2x+2}\text{dx}}} \int{\arcsin \sqrt{\frac{a-x}{a+x}}\text{dx}} \int{\frac{x\mathrm{d}x}{\left( 1-x^3 \right) \sqrt{1-x^2}}} \int{\frac{1}{x}\sqrt{\frac{1+x}{1-x}}\text{d}x} \int{\frac{x^3\mathrm{d}x}{x^4+x^3+x+1}} \int{\frac{\mathrm{d}x}{\sqrt{1-x}+\sqrt{1+x^2}}} \int{\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}}\text{d}x \int{\frac{\sqrt{1-x^4}}{1+x^4}\mathrm{d}x} \int{\sqrt{x+2+\sqrt{\left( x+1 \right) \left( x+3 \right)}}\mathrm{d}x} \int{\frac{x+\sin x}{1+\cos x}\text{dx}} \int{\frac{x\cos x\mathrm{d}x}{\left( x+\cos x \right) ^2}} \int{\frac{1}{\cos x\sqrt{\cos 2x}}\text{d}x} \int{xe^x\sin x\cos x\mathrm{d}x} \int{e^x\frac{1+\sin x}{1+\cos x}\text{d}x} \color{blue}{\int{\ln x^2\arcsin x}\text{d}x } \int{\frac{\sin \left( \ln x \right)}{x^2}\text{d}x} {\int{\frac{\sin x\text{d}x}{\sqrt{1+\sin 2x}}} } \int{\frac{\sqrt{\cos 2x}}{\sin x}\text{d}x} \color{blue}{\int{\frac{x\ln x}{\left( x^2+1 \right) ^{\frac{3}{2}}}\text{d}x} } \int{\frac{\sin x\cos x}{\sin ^4x+\cos ^4x}\text{dx}} \color{blue}{\int{\frac{x^2}{\left( x\sin x+\cos x \right) ^2}\text{d}x}} \color{blue}{\int{\frac{2x+\sin 2x}{\left( \cos x-x\sin x \right) ^2}\text{d}x} } \int{\frac{1-x}{2x+x^2e^{-x}}\mathrm{d}x} \color{blue}{\int{\frac{1}{\sqrt{2}+\sqrt{1-x}+\sqrt{1+x}}\text{d}x} } \int{\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\text{d}x} \color{blue}{\int{\frac{x-2}{\sqrt{e^x-x^2}}\text{d}x} } \int{\frac{e^x\left( \left( x-1 \right) ^2-2 \right)}{\left( x^2-1 \right) ^2}\text{d}x} \int{e^x\left[ \frac{1}{\sqrt{1+x^2}}+\frac{1-2x^2}{\left( \sqrt{1+x^2} \right) ^5} \right]}\mathrm{d}x \int{\frac{1}{\sqrt{\text{th}x+1}}\text{d}x} \int{\sqrt{\frac{x^2}{2}-\frac{2}{x^2}}\text{d}x} \int{\mathrm{arc}\sin ^nx\mathrm{d}x} \int{\arctan \left( 1-\frac{1}{x} \right)}\text{d}x \int{\frac{\text{d}x}{a^2\sin ^2x+b^2\cos ^2x}} \int{\sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x...}}}}\text{d}x} \int{\frac{\mathrm{d}x}{x\sqrt{1+x^{1011}+x^{2022}}}} \int{\frac{\mathrm{d}x}{\sqrt{x^2+x^n}}} \int{ \frac{\displaystyle\sum_{k=0}^{n}{\left( k+1 \right) x^k}}{\displaystyle\sum_{k=0}^{n+1}{x^k}}\text{d}x} \int{\frac{x^n}{\displaystyle\sum_{k=0}^n{\frac{x^k}{k!}}}\text{d}x} 3. 正文·百题集锦王者不需要热身！\color{blue}{\int{\frac{\cos \left( \sin t \right) +\cos ^2t}{1+\sin t\sin \left( \sin t \right)}\text{d}t} } \int{\frac{x^2-2\sinh ^2x}{\left( x^2-1 \right) \sinh \left( 2x \right) +2x}}\text{d}x \int{\left[ \frac{x\sin \left( \ln \left( x+1 \right) \right)}{\left( 1+\sqrt{1+x} \right)} \right] ^2\text{d}x} \int{\frac{x+1}{\exp \left\{ 2/x \right\}}\exp \left\{ xe^{-\frac{1}{x}} \right\} \text{d}x} \int{\frac{\ln \left( \sqrt{1+x^2}-x \right) \text{d}x}{\sqrt{1+x^2}\sqrt{x+\sqrt{1+x^2}}}} \color{blue}{\int{\frac{x^2\text{d}x}{\left( x^2-4 \right) \sin x+4x\cos x}} } \int{\frac{x^3-2}{\left( x^3+1 \right) ^2}\sqrt{x^3-x^2+1}\text{d}x} \int{\frac{\sin ^3x}{\sin ^3x+\cos ^3x}\text{d}x} \int{\left( x^m+x^{2m}+x^{3m} \right) \sqrt[m]{2x^{2m}+3x^m+6}\mathrm{d}x} \color{blue}{\int{\frac{\cos 2x-\tan x\cdot \cot \left( \tan x \right)}{\sin 2x-\tan \left( \tan x \right) \ln \left( \cos ^2x \right)}\text{d}x} } \int{\frac{\text{dx}}{\sin ^6x+\cos ^6x}} \int{\frac{\sin 4x}{\sin ^8x+\cos ^8x}\text{d}x} \color{blue}{\int{\frac{\cos x\text{d}x}{\sqrt{2+\sin 2x}}}} \int{\frac{\text{d}x}{\left( x-a \right) ^4+\left( x-b \right) ^4}} \int{\sqrt{x+a^2\sqrt{x-a}}}\text{d}x \int{\frac{x^2\text{d}x}{a^2+x^2+\sqrt{a^2+x^2}}} \int{\frac{x+x^3+x^5+2x^7}{\exp \left\{ x^2-x^4 \right\}}\text{d}x} \int{\frac{x\cos x\text{d}x}{3+4\sin x-\cos 2x}} \int{\frac{x\sin x+\cos x}{\left( x+\cos x \right) ^2}\text{d}x} \color{blue}{\int{\frac{\left( \sin x+1 \right) ^2+x\cos x}{\left( x\sin x-\cos x \right) ^2}\text{d}x} } \int{\frac{x^2+2\cos x-1}{\left( 1+x^2 \right) \sin x-2x}\mathrm{d}x} \int{\frac{x\left( 1+x \right) \text{d}x}{\left( e^x+x+1 \right) ^2}} \color{blue}{\int{\frac{\tan x-\cos x}{\left( 2+e^{\sin x}\cos x \right) ^2}\text{d}x} } \int{\frac{\left( 2\cosh x-1 \right) \text{d}x}{\left( e^x\sin x+\cos x \right) \left( \cos x-e^{-x}\sin x \right)}} \int{e^{-x^2}\left( \left( x^2+2^{-1} \right) ^{-2}-2 \right)}\text{d}x \int{\frac{\cot x-\sec ^2x+1}{\cot x-1}\left( \frac{e^x}{\tan ^{2020}x} \right) ^{\frac{1}{2021}}\mathrm{d}x} \int{\left( \frac{x^2-3x+1/3}{x^3-x+1} \right) ^2\mathrm{d}x} \color{blue}{\int{\frac{\text{d}x}{\left( 1+x\tan x \right) ^2}}}\int{\frac{\text{d}x}{\left( \sin x+a\sec x \right) ^2}} \color{blue}{\int{\frac{\cos x\left( 2020x+2019\sin x\cos x \right)}{\left( x\sin x+\cos x \right) ^3}\text{d}x} }\int{\frac{x^4\cos ^3x-x\sin x+\cos x}{e^{-x\sin x-\cos x}x^2\cos ^2x}}\text{d}x \color{blue}{\int{\frac{\left( x+1 \right) e^x}{\sqrt{a^2-e^x}}\text{d}x}} \int{\sqrt{\tan x}\text{dx}} \int{\sqrt{\tan x+1}\text{dx}} \int{\sqrt{\frac{1+\sqrt{2}\sin x}{1+\sqrt{2}\cos x}}\text{d}x} \int{\sqrt[3]{\frac{1+\sin x}{1-\sin x}}\text{d}x} \int{\frac{1}{\sqrt{\tan x}}\text{d}x} \int{\frac{1}{\sqrt{\tan x+1}}\text{d}x} \color{blue}{\int{\frac{\text{d}x}{\left( 1+\sin x \right) ^n}} } \int \left(\frac{\arctan x}{\arctan x-x}\right)^2 \text{dx} \int{\frac{2x\arctan x-1}{\arctan ^2x}\text{d}x} \int{\mathrm{arc}\cot \left( x^2+x+1 \right)}\mathrm{d}x \color{blue}{\int{\text{arctan}\frac{x^3-x^2-4x-1}{x^3+4x^2+x-1}\text{d}x}} \int{\arccos \left( 7x^2-\sqrt{49x^4+1-50x^2} \right) \ln x\text{d}x} \color{blue}{\int{\frac{\sqrt{1+x^4}}{1-x^4}}\mathrm{d}x } \color{blue}{\int{\frac{x^4-1}{x^8+1}\sqrt{1+x^4}\text{d}x} } \int{\frac{\arcsin \sqrt{x}\arccos \sqrt{x}}{\sqrt{1-x}}}\text{d}x \int{\frac{\mathrm{arc}\tan x}{\left( x+1/x \right) ^2}\mathrm{d}x} \int{\frac{2x\arctan x\left( \arctan x+x \right)}{\left( 1+x^2 \right) ^2}\text{d}x} \int{\frac{x}{\sqrt{1-x^2}}}\ln \frac{x}{\sqrt{1-x^2}}\text{d}x \int{\frac{ \ln \left( x+m \right) - \ln \left( x+n \right)}{\left( x+m \right) ^2\left( x+n \right) ^2}\text{d}x} \int{\frac{\left( x+m \right) \ln \left( x+m \right) +\left( x+n \right) \ln \left( x+n \right)}{\left( x+m \right) ^2\left( x+n \right) ^2}\text{d}x} \int{\frac{1}{\ln x}\left( \frac{1}{\ln ^2x}-\frac{1}{2} \right)}dx \color{blue}{\int{\frac{\frac{\ln x}{x}+\ln ^2x}{e^{-2x}+\ln ^2x}\text{d}x} } \int{\frac{\ln ^2x+1-\left( \ln ^2x+1 \right) ^{-\frac{1}{2}}}{2x\ln x}}\text{dx} \int{\frac{\ln \left( x^2 \right) +1-\left( \ln \left( x^2 \right) +1 \right) ^{-\frac{1}{2}}}{2x\ln x}}\text{dx} \int{\frac{\sqrt{1+\sqrt{1+x^4}+\sqrt{2}\sqrt{1+x^4+\sqrt{1+x^4}}}}{1+x^4}\mathrm{d}x} \int{\csc ^2x\ln \left( \cos x+\sqrt{\cos 2x} \right) \text{d}x} \color{blue}{\int{\sqrt{\frac{x+\sqrt{x}}{x-\sqrt{x}}}}\text{d}x } \int{\frac{\left( 1+x \right) \sin x\mathrm{d}x}{\left( x^2+2x \right) \cos ^2x-\left( 1+x \right) \sin 2x}} \int{\frac{\left( x-1 \right) \text{d}x}{\left( \sqrt{x}+x+x\sqrt{x} \right) \sqrt{\left( x+1 \right) \sqrt{x}}}} \int{\sqrt[3]{x\left( 1-x \right)(1+x)}}\text{d}x \int{\left( \frac{1}{x}-\sqrt{x} \right) ^{-\frac{1}{2}}\text{d}x} \int{\frac{\sin 3x\mathrm{d}x}{3\sin x+\sin ^3x}} \int{{\sqrt{\frac{1+x^2+\sqrt{1+x^2+x^4}}{1+x^2-\sqrt{1+x^2+x^4}}}}\text{d}x } \int{\tan x\tan 2x\tan 3x\mathrm{d}x} \int{\sqrt{\frac{x}{1-x}\mathrm{arc}\sin ^4\sqrt{x}}\mathrm{d}x} \int{\sqrt{\frac{\sin \left( x+\xi \right)}{\sin \left( x-\xi \right)}}}\text{d}x \int{\sqrt{\frac{\left( 1-\sin x \right) \left( 2-\sin x \right)}{\left( 1+\sin x \right) \left( 2+\sin x \right)}}}\text{d}x \int{\frac{1+\sec \left( x \right) \sec \left( \sec x \right) \csc \left( \sec x \right)}{\sqrt{\cos ^2x+\left( \csc x-\sin x \right) ^2\csc ^2\left( \sec x \right)}}}\mathrm{d}x \int{\left( x^{x^2+2}+1 \right) \left( \ln x^2+1 \right) \frac{\text{d}x}{x}} \color{blue}{\int{\frac{\text{d}x}{\sqrt{\left( x+x^{-1} \right) ^2-12}}} } \color{blue}{\int{\frac{x\left( 2-x \right) e^x\cos 2x+e^{2x}-x^4}{\left( e^x\cos x+x^2\sin x \right) \sqrt{x^4-e^{2x}}}\frac{\text{d}x}{\sqrt{\cos 2x}}} } \int{\exp \left\{ \sec x \right\} \frac{\left( 1+\sin x \right) \left( 1+\tan x \right)}{\cos ^2x}\text{d}x} \int{\frac{x}{\sqrt{e^{-x}-e^{-2x}}}\text{d}x} \int{\frac{\text{d}x}{\sqrt[3]{e^{-2x}+e^{-3x}}-\sqrt{e^{-x}+e^{-2x}}}} \int{\frac{\mathrm{d}s}{\lambda +\sqrt{1-x^2}}} （其中 \frac{s}{x}=t^2+1 ， t=\frac{\sqrt{1-x^2}-\lambda}{\sqrt{1-\lambda ^2}-x} ）\int{\frac{\mathrm{d}x}{\sqrt{e^{2x}-x^2}}\left[ 1-x\frac{e^x+1}{e^x+x} \right]} \color{blue}{\int{\frac{x\left( x^2+x\tan x+1 \right)}{\left( x\tan x-1 \right) ^2}\text{d}x} } \color{blue}{\int{\frac{x\left( x^2+x\tan x+1 \right)}{\left( x\tan x+1 \right) ^2}\text{d}x} } \int{\frac{\sec ^2x}{\left( \sec x+\tan x \right) ^n}\text{d}x} \int{\sqrt{\csc ^2x+\cot ^2x}\text{d}x} \color{blue}{\int{\frac{x\text{d}x}{2\left( \sec x+\tan x \right) -\cos x}} } \int{\frac{\text{d}x}{\sin x+\cos x+\sin x\cos x}} \int{\frac{\text{d}x}{\cos \left( x-1 \right) \cos \left( x-2 \right) \cos \left( x-3 \right)}} \int{\frac{\left( x\cos x-\sin x \right) \text{d}x}{\left( x+a\sin x \right) \left( x+b\sin x \right)}} \int{\frac{\tan ^2x}{1+\sec ^2x}\sqrt[3]{\tan x\sec x}\mathrm{d}x} \int{\frac{\sqrt[3]{\cos x}}{3+\sin x}\mathrm{d}x} \int{\frac{\text{d}x}{\sqrt{\sin 2x\cos ^2x}}} \int{\frac{\sin ^2\frac{x}{2}\tan \frac{x}{2}}{\sqrt{\cos x+\cos ^2x+\cos ^3x}}\text{d}x} \int{\frac{\sec x}{\sqrt{\sin \left( 2x+\xi \right) +\sin \xi}}\text{d}x} \int{\frac{\sin \left( \text{arccot} x \right)}{\cot \left( \arcsin x \right)}\text{d}x} \int{\sqrt{\frac{\csc x-\cot x}{\csc x+\cot x}}\frac{\sec x}{\sqrt{1+\text{2}\sec x}}\text{d}x} \int{\sqrt[3]{\frac{1+\sin x}{5-4\sin x}}\mathrm{d}x} \int{e^x\frac{2x+\left( x+1 \right) \sin 2x}{1+\cos 2x}\text{d}x} \int{\frac{\text{d}x}{\left( x^2-x+2 \right) \sqrt{x^2+x+1}}} \color{blue}{\int{\frac{x+x^{-1}+1}{\sqrt{\left( x^2+x-1/3 \right) ^2+4x/3}}\text{d}x} } \color{blue}{\int{\frac{x^n\text{d}x}{\sqrt{\sum_{k=0}^n{\left( k+1 \right) x^k}}}} } \color{blue}{\int{\frac{2n!\sin x+x^n}{e^x+\sin x+\cos x+\displaystyle\sum_{k=0}^n{\frac{x^k}{k!}}}}\text{d}x } 设 J_n\left( x \right) =\frac{1}{\pi}\int_0^{\pi}{\cos \left(nt- x\sin t \right) \text{d}t} ，(结果用J_n(x)表示)，求 \color{gray}{\int{\sin(x)J_0\left( x\right) \text{d}x} }附注：知乎上有很多大佬的相关解答，搜索“百题解答”即可。 |
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