«高数下基本计算练习»
by pluvet on Jan 3, 2020

求导总结

  1. $y=\int_{0}^{1+\sin t}\left(1+e^{\frac{1}{u}}\right) d u$ , $\left\{\begin{array}{l}{x=\cos 2 v} \\ {t=\sin v}\end{array} \quad\left(0<v<\frac{\pi}{2}\right)\right.$ 求 $\frac{\mathrm{d} y}{\mathrm{d} x}$
  2. $x^{3}+3 x^{2} y-2 y^{3}=2$ 确定 $y=y(x)$, 求其极值
  1. $\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} y}{\mathrm{d} t} \cdot \frac{\mathrm{d} t}{\mathrm{d} x}=-\frac{\cos t}{4 t}\left(1+\mathrm{e}^{\frac{1}{1+\sin t}}\right)$

积分总结

  1. $\int \sqrt{\frac{1}{1+2x^2}}dx$ https://www.pluvet.com/archives/384.html
  2. $\int \sqrt{\frac{1}{x^2}-1}\:d\:x$ https://www.pluvet.com/archives/384.html
  3. $\int\sqrt{1-x^2}dx$
  4. $\int x\sqrt{1-x^2}dx$
  5. $\int \frac{1}{\sqrt{\frac{1}{x^2}-1}}\:d\:x$
  6. $\lim _{n \to\infty} \frac{\sum_{k=1}^{n} \sqrt{n^{2}-k^{2}}}{n^{2}}$https://www.pluvet.com/archives/385.html
  7. $\lim\limits_{n \to \infty}\sum\limits_{k=0}^n \dfrac{\sqrt{n}}{n+k^2}(n=1,2,\cdots)$ https://www.pluvet.com/archives/385.html
  8. $\int \frac{1}{\cos (x)+1} d x$ https://www.pluvet.com/archives/370.html
  9. $\int \frac{1}{\cos (x)-1} d x$ https://www.pluvet.com/archives/370.html
  10. $\int \sqrt{1+\cos x} d x$ https://www.pluvet.com/archives/370.html
  11. $\int _0^{+\infty }\frac{\ln \left(x\right)}{1+x^2}\mathrm{\:}dx$ https://www.pluvet.com/archives/367.html
  12. $\int ^1_{-1}\sqrt{\frac{1+x}{1-x}}\rm{d}x$ https://www.pluvet.com/archives/366.html
  13. $\int\sec^3x\:\rm{d}x$ https://www.pluvet.com/archives/364.html
  14. $\int \frac{1}{x\left(\ln \left(x\right)\right)^2}dx$

微分方程总结

  1. $4 x \mathrm{d} x-3 y \mathrm{d} y=3 x^{2} y \mathrm{d} y$
  1. 分离变量

极限总结

默写:

  1. $e^x$
  2. $\sin x$
  3. $\cos x$
  4. $\arctan x$
  5. $\ln(x+1)$
  6. $\frac{1}{1-x}$
  7. $\frac{1}{1+x}$
  8. $(1+x)^{\alpha}$
  9. $\tan x$
  10. $\arcsin x$

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