«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. $\int \frac{1}{x\left(\ln \left(x\right)\right)^2}dx$
• $\int \frac{\cos x}{1-\sin x} d x$
• $\int_{1}^{4} \sqrt{y} \ln y d y$
• $\int \frac{t}{t^{4}+2} d t$
• $\int_{-1}^{1} \frac{e^{\arctan y}}{1+y^{2}} d y$
• $\int_{2}^{4} \frac{x+2}{x^{2}+3 x-4} d x$
• $\int_{0}^{1}(3 x+1)^{\sqrt{2}} d x$
• $\int \frac{\sin ^{3} x}{\cos x} d x$
• $\int_{0}^{1} \frac{x}{(2 x+1)^{3}} d x$
• $\int t \sin t \cos t d t$
• $\int \frac{\cos (1 / x)}{x^{3}} d x$
• $\int \frac{1}{x^{3} \sqrt{x^{2}-1}} d x$
• $\int \sin ^{5} t \cos ^{4} t d t$
• $\int x \sec x \tan x d x$
• $\int_{0}^{\pi} t \cos ^{2} t d t$
• $\int e^{x+e^{x}} d x$
• $\int \arctan \sqrt{x} d x$
• $\int \frac{2 x-3}{x^{3}+3 x} d x$
• $\int \ln \left(1+x^{2}\right) d x$
• $\int_{0}^{\sqrt{2} / 2} \frac{x^{2}}{\sqrt{1-x^{2}}} d x$
• $\int_{1}^{4} \frac{e^{\sqrt{t}}}{\sqrt{t}} d t$
• $\int e^{2} d x$
• $\int \frac{\ln x}{x \sqrt{1+(\ln x)^{2}}} d x$
• $\int_{0}^{1}(1+\sqrt{x})^{s} d x$
• $\int_{0}^{1} \frac{1+12 t}{1+3 t} d t$
• $\int \frac{d x}{1+e^{x}}$
• $\int \ln (x+\sqrt{x^{2}-1}) d x$
• $\int \sqrt{\frac{1+x}{1-x}} d x$
• $\int \sqrt{3-2 x-x^{2}} d x$
• $\int_{-\pi / 2}^{\pi / 2} \frac{x}{1+\cos ^{2} x} d x$
• $\int_{0}^{\pi / 4} \tan ^{3} \theta \sec ^{2} \theta d \theta$
• $\int \frac{\sec \theta \tan \theta}{\sec ^{2} \theta-\sec \theta} d \theta$
• $\int \theta \tan ^{2} \theta d \theta$
• $\int \frac{\sqrt{x}}{1+x^{3}} d x$
• $\int x^{5} e^{-x^{3}} d x$
• $\int x^{3}(x-1)^{-4} d x$
• $\int \frac{1}{x \sqrt{4 x+1}} d x$
• $\int \frac{1}{x \sqrt{4 x^{2}+1}} d x$
• $\int x^{2} \sinh m x d x$
• $\int \frac{d x}{x+x \sqrt{x}}$
• $\int x \sqrt[3]{x+c} d x$
• $\int(1+\tan x)^{2} \sec x d x$
• $\int_{0}^{1} \frac{3 x^{2}+1}{x^{3}+x^{2}+x+1} d x$
• $\int \sin \sqrt{a t} d t$
• $\int_{-1}^{2}\left|e^{x}-1\right| d x$
• $\int_{1}^{3} \frac{e^{3 / x}}{x^{2}} d x$
• $\int_{x / 4}^{x / 2} \frac{1+4 \cot x}{4-\cot x} d x$
• $\int \frac{1+\sin x}{1+\cos x} d x$
• $\int_{\pi / 6}^{\pi / 3} \frac{\sin \theta \cot \theta}{\sec \theta} d \theta$
• $\int_{0}^{\pi} \sin 6 x \cos 3 x d x$
• $\int \frac{\tan ^{-1} x}{x^{2}} d x$
• $\int \sqrt{1+e^{x}} d x$
• $\int \frac{(x-1) e^{x}}{x^{2}} d x$
• $\int_{0}^{1} x \sqrt{2-\sqrt{1-x^{2}}} d x$
• $\int \frac{1}{x^{2} \sqrt{4 x+1}} d x$
• $\int \frac{d x}{x\left(x^{4}+1\right)}$
• $\int(x+\sin x)^{2} d x$
• $\int \frac{d x}{\sqrt{x}+x \sqrt{x}}$
• $\int \frac{x \ln x}{\sqrt{x^{2}-1}} d x$
• $\int \frac{d x}{x^{4}-16}$
• $\int \frac{d \theta}{1+\cos \theta}$
• $\int \sqrt{x} e^{\sqrt{x}} d x$
• $\int \frac{\sin 2 x}{1+\cos ^{4} x} d x$
• $\int \frac{1}{\sqrt{x+1}+\sqrt{x}} d x$
• $\int_{1}^{\sqrt{3}} \frac{\sqrt{1+x^{2}}}{x^{2}} d x$
• $\int \frac{e^{2 x}}{1+e^{x}} d x$
• $\int \frac{x+\arcsin x}{\sqrt{1-x^{2}}} d x$
• $\int \frac{d x}{x \ln x-x}$
• $\int \frac{x e^{x}}{\sqrt{1+e^{x}}} d x$
• $\int x \sin ^{2} x \cos x d x$
• $\int \sqrt{1-\sin x} d x$
• $\int \frac{d x}{x^{2} \sqrt{4 x^{2}-1}}$
• $\int \frac{d \theta}{1+\cos ^{2} \theta}$
• $\int \frac{1}{\sqrt{\sqrt{x}+1}} d x$
• $\int_{x / 4}^{\pi / 3} \frac{\ln (\tan x)}{\sin x \cos x} d x$
• $\int \frac{x^{2}}{x^{6}+3 x^{3}+2} d x$
• $\int \frac{1}{1+2 e^{x}-e^{-x}} d x$
• $\int \frac{\ln (x+1)}{x^{2}} d x$
• $\int \frac{4^{x}+10^{x}}{2^{x}} d x$
• $\int \frac{x^{2}}{\sqrt{x^{2}+1}} d x$
• $\int \frac{1+\sin x}{1-\sin x} d x$
• $\int \frac{\sec x \cos 2 x}{\sin x+\sec x} d x$
• $\int \frac{\sin x \cos x}{\sin ^{4} x+\cos ^{4} x} d x$
• $\int_{2}^{\infty} e^{-5 p} d p$
• $\int_{0}^{\infty} \frac{x^{2}}{\sqrt{1+x^{3}}} d x$
• $\int_{-\infty}^{\infty} x e^{-x^{2}} d x$
• $\int_{0}^{\infty} \sin ^{2} \alpha d \alpha$
• $\int_{1}^{\infty} \frac{1}{x^{2}+x} d x$
• $\int_{-\infty}^{0} z e^{2 z} d z$
• $\int_{1}^{\infty} \frac{\ln x}{x} d x$
• $\int_{-\infty}^{0} \frac{z}{z^{4}+4} d z$
• $\int_{0}^{\infty} e^{-\sqrt{y}} d y$
• $\int_{0}^{1} \frac{1}{x} d x$
• $\int_{-2}^{14} \frac{d x}{\sqrt[4]{x+2}}$
• $\int_{-2}^{3} \frac{1}{x^{4}} d x$
• $\int_{-\infty}^{0} 2^{r} d r$
• $\int_{-\infty}^{\infty}\left(y^{3}-3 y^{2}\right) d y$
• $\int_{1}^{\infty} \frac{e^{-1 / x}}{x^{2}} d x$
• $\int_{0}^{\infty} \sin \theta e^{\cos \theta} d \theta$
• $\int_{2}^{\infty} \frac{d v}{v^{2}+2 v-3}$
• $\int_{2}^{\infty} y e^{-3 y} d y$
• $\int_{1}^{\infty} \frac{\ln x}{x^{2}} d x$
• $\int_{e}^{\infty} \frac{1}{x(\ln x)^{2}} d x$
• $\int_{1}^{\infty} \frac{d x}{\sqrt{x}+x \sqrt{x}}$
• $\int_{0}^{5} \frac{1}{\sqrt[3]{5-x}} d x$
• $\int_{-1}^{2} \frac{x}{(x+1)^{2}} d x$
• $\int_{0}^{1} \frac{d x}{\sqrt{1-x^{2}}}$

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

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. $\arcsin x$