抽象解法合集

裂项

$$\begin{array}{rl}\int \frac{1+x^2}{1+x^4}dx& =\int \frac{1+x^2}{(1+x^2{)}^2-2x^2}\\ & =\int \frac{1}{2+2x^2-2\sqrt{2}x}+\frac{1}{2+2x^2+2\sqrt{2}x}dx\\ & =\int \frac{1}{(\sqrt{2}x-1{)}^2+1}+\frac{1}{(\sqrt{2}x+1{)}^2+1}dx\\ & =\frac{1}{\sqrt{2}}(\arctan (\sqrt{2}x-1)+\arctan (\sqrt{2}x+1))+C\\ & =\frac{1}{\sqrt{2}}\arctan (\frac{2x^2}{2-2x^2})+C\end{array}$$

$$\begin{array}{rl}{f}^{\prime }(0)& =\lim \frac{f(x)-f(0)}{x}\\ & =\lim \frac{g(x)-g(0)}{x^2}\\ & =\lim \frac{g^{\prime }(\xi )-g^{\prime }(0)}{x}\\ & =\lim \frac{g^{\prime }(\xi )-g^{\prime }(0)}{\xi }\frac{\xi }{x}\\ & =g^{\prime \prime }(0)\lim \frac{\xi }{x}\end{array}$$

嗯凑

$$\begin{array}{rl}\int \frac{\cos x}{\cos x+2\sin x}dx& =\int \frac{\cos x-1/2\cos x-\sin x}{\cos x+2\sin x}+\frac{1}{2}dx\\ & =\int \frac{\frac{1}{2}d(\cos x+2\sin x)}{\cos x+2\sin x}+\frac{1}{2}x\\ & =\frac{1}{2}(\ln (\cos x+2\sin x)+x)+C\end{array}$$

指数带拖油瓶

$$\begin{array}{rl}\int \frac{1+2x}{x^2}{e}^{-2x}dx& =-\int d\frac{e^{-2x}}{x}\end{array}$$