积分表(1)

这大概算是真正的第一篇笔记了。为应对之后笔记中大量的公式需求，先推理一遍同济大学高等数学积分表，以熟悉LaTeX的使用。

（一）含有$ax+b$的积分

  $1.\int\dfrac{dx}{ax+b}=\dfrac{1}{a}\ln\left|ax+b\right|+C$

$$\int\dfrac{dx}{ax+b}=\int\dfrac{1}{a}\dfrac{d\left(ax+b\right)}{ax+b}=\dfrac{1}{a}\ln\left|ax+b\right|+C$$

  $2.\int\left(ax+b\right)^{\mu}dx=\dfrac{1}{a\left(\mu+1\right)}\left(ax+b\right)^{\mu+1}+C\quad\left(\mu\neq-1\right)$

$$\int\left(ax+b\right)^{\mu}dx=\int\left(ax+b\right)^{\mu}\dfrac{d\left(ax+b\right)}{a}=\dfrac{1}{a\left(\mu+1\right)}\left(ax+b\right)^{\mu+1}+C$$

由于$\mu=-1$时，原式变为积分表1式，因此$\mu\neq-1$。

  $3.\int\dfrac{x}{ax+b}dx=\dfrac{1}{a^2}\left(ax+b-b\ln\left|ax+b\right|\right)+C$

$$\int\dfrac{x}{ax+b}dx=\int\dfrac{1}{a}\dfrac{x+\dfrac{b}{a}-\dfrac{b}{a}}{x+\dfrac{b}{a}}dx=\int\dfrac{1}{a}\left(1-\dfrac{b}{ax+b}\right)\dfrac{1}{a}d\left(ax+b\right)=\dfrac{1}{a^2}\left(ax+b-b\ln\left|ax+b\right|\right)+C$$

  $4.\int\dfrac{x^2}{ax+b}dx=\dfrac{1}{a^3}\left[\dfrac{1}{2}\left(ax+b\right)^2-2b\left(ax+b\right)+b^2\ln\left|ax+b\right|\right]+C$

$$\int\dfrac{x^2}{ax+b}dx=\dfrac{1}{a^2}\int\left[\left(ax+b\right)-\dfrac{2abx}{ax+b}-\dfrac{b^2}{ax+b}\right]dx=\dfrac{1}{a^3}\left[\dfrac{1}{2}\left(ax+b\right)^2-2b\left(ax+b\right)+b^2\ln\left|ax+b\right|\right]+C$$