一些常见的积分

凑微分类

$${\int }_0^{+\infty }{e}^{-x}dx=-e^{-t}{|}_0^{+\infty }=e^{-0}-e^{{}^{-\infty }}=1所以{\int }_0^{+\infty }{e}^{-\lambda x}dx=\frac{1}{\lambda }$$

$$\int \sqrt{a^2-r^2}rdr=\frac{1}{2}\int \sqrt{a^2-r^2}d{r}^2=-\frac{1}{2}\int \sqrt{a^2-r^2}d(a^2-r^2)=-\frac{1}{2}\frac{2}{3}(a^2-r^2{)}^{\frac{3}{2}}+C$$

$$\int \frac{1-x^2}{1+x^4}dx=\int \frac{\frac{1}{x^2}-1}{\frac{1}{x^2}+x^2}dx=\int \frac{d(-\frac{1}{x}-x)}{(-\frac{1}{x}-x{)}^2+2}$$

分布积分类

$$\int lnx\ast x^{2}dx=lnx\ast \frac{x^3}{3}-\int \frac{1}{x}\frac{x^3}{3}dx=lnx\ast \frac{x^3}{3}-\frac{1}{3}\frac{x^3}{3}+C$$
$$\int x{e}^{x}dx=(x-1)e^x+C$$
$$\int x^2{e}^{x}dx=(x^2-2x+2)e^x+C$$
$$\int x^3{e}^{x}dx=(x^3-3x^2+6x-6)e^x+C$$

其他

$$\begin{gathered} \int \sqrt{a^2-x^2}da,x=asint,dx=acost \\ {a}^2\int co{s}^{2}tdt=\frac{a^2}{2}\int (1+cos2y)dy \\ =\frac{a^2}{2}(y+\frac{1}{2}\int (cos2t)d(2t))==\frac{a^2}{2}(y+sin2t) \end{gathered}$$

其他

$$\begin{gathered} \int \frac{1}{a^2-x^2}dx直接分解，或者三角代换 \\ =\int \frac{1}{2a}(\frac{1}{a-x}+\frac{1}{a+x})dx=\frac{1}{2a}(ln|x+a|+ln|a-x|)=\frac{1}{2a}ln|\frac{x+a}{x-a}|+c \end{gathered}$$

其他

$$\int \frac{1}{a{x}^2+bx+c}dx不能分解时加常数,凑arctanx$$
$$\int \frac{x}{1+x^3}dx=\int \frac{x}{(1+x)(1-x+x^2)}dx$$

其他

$$\begin{gathered} 一次方和三次方比较好化简\int \frac{x^3}{1+x^4}dx=\frac{1}{4}\int \frac{d{x}^4}{1+x^4} \\ \int \frac{x}{1+x^4}=\frac{1}{2}\int \frac{d{x}^2}{1+(x^2{)}^2} \end{gathered}$$

$$A=\int \frac{x^2+1}{1+x^4}dx=\frac{1}{\sqrt{2}}arctan(\frac{x^2-1}{\sqrt{2}x})+C$$
$$B=\int \frac{x^2-1}{1+x^4}dx=\frac{1}{2\sqrt{2}}ln|\frac{x^2-\sqrt{2}x+1}{x^2+\sqrt{2}x+1}|+C$$
$$\int \frac{1}{1+x^4}用组合积分法=\frac{1}{2}(A-B)$$

"定"积分与无穷级数

$$\begin{gathered} \sum _1^{n}f(\frac{i}{n})\ast \frac{1}{n}={\int }_0^{1}f(x)dx \\ 比如li{m}_{n\to +\infty }\sum _1^n\frac{1}{i+n}=\sum _1^n\frac{1}{\frac{1}{n}+1}\ast \frac{1}{n}={\int }_0^1\frac{1}{x+1}dx \end{gathered}$$

定积分

$${\int }_0^{\pi }xf(sinx)dx=\frac{\pi }{2}{\int }_0^{\pi }f(sinx)dx(用区间再现构造方程直接解出目标积分呢)$$

$$\int 0到\frac{\pi }{2}华理式公式$$

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$$\begin{gathered} 计算I={\int }_0^{\pi }xsi{n}^{n}xdx \\ {\int }_0^{\pi }xf(sinx)dx=\frac{\pi }{2}{\int }_0^{\pi }f(sinx)dx \end{gathered}$$

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$$\begin{gathered} [0,\pi ],si{n}^n(x)总是关于x=\frac{\pi }{2}对称 \\ {\int }_0^{\pi }si{n}^{n}xdx=2{\int }_0^{\frac{\pi }{2}}si{n}^{n}xdx \\ {\int }_0^{\pi }co{s}^{n}xdx=\{\begin{array}{l}0,n为奇数\\ 2{\int }_0^{\frac{\pi }{2}}co{s}^{n}xdx,n为偶数\end{array} \end{gathered}$$