第二类换元法之其它方法

1，双曲代换：双曲代换类似于三角代换，实际上，能够应用三角代换的地方，大部分都能应用双曲代换。因为双曲函数也有平方和公式。

$x=\sinh t，dx=\cosh tdt$

$x=\cosh t, dx=\sinh tdt$

$\tanh x=\frac{\sinh x}{\cosh x}, \coth x=\frac{\cosh x}{\sinh x}$

$\cosh^2x=\sinh^2x+1$

\begin{align*} \int \frac{dx}{\sqrt{x^2-a^2}}&=\int \frac{a\sin htdt}{\sqrt{a^2\cosh^2t-a^2}}\\ &=\int \frac{\sin ht}{\sin ht}dt=\int dt=t+C\end{align*}

$t=\cos h^{-1} \frac{x}{a}=\ln\left(\frac{x}{a}+\sqrt{\left(\frac{x}{a}\right)^2-1}\right)$

\begin{align*}\int \frac{dx}{\sqrt{x^2-a^2}}&=\ln\left(\frac{x}{a}+\sqrt{\left(\frac{x}{a}\right)^2-1}\right)+C\\ & =\ln\left(\frac{x}{a}+\frac{1}{a}\sqrt{x^2-a^2}\right)+C\\ & =\ln\left(x+\sqrt{x^2-a^2}\right)-\ln a+C\\ & =\ln\left(x+\sqrt{x^2-a^2}\right)+C\end{align*}

\begin{align*}\int\frac{dx}{\sqrt {x^2+a^2}}&=\int \frac{a\cosh tdt}{\sqrt {a^2\sinh^2t+a^2}} =\int dt\\ &=t+C =\sinh^{-1} \frac{x}{a}+C\\ &=\ln|x+\sqrt{x^2+a^2}|+C\end{align*}

2，例代换：另一个常用的第二类换元法是倒代换，就是做代换 $$\displaystyle\quad x=\frac{1}{t}, dx=-\frac{1}{t^2}dt$$

$\int \frac{\sqrt{a^2-x^2}}{x^4}dx=\int \frac{\sqrt{a^2-\frac{1}{t^2}}}{\frac{1}{t^4}}\cdot \left(-\frac{1}{t^2}\right) d=-\int t^2\cdot \sqrt{a^2-\frac{1}{t^2}}dt$

\begin{align*}\int \frac{\sqrt{a^2-x^2}}{x^4}dx&=-\int t\sqrt{a^2t^2-1} dt=-\frac{1}{2a^2}\int \sqrt{u}du\\ &=-\frac{1}{2a^2}\cdot \frac{2}{3}u^{\frac{3}{2}}+C=-\frac{1}{3a^2}\left(a^2t^2-1\right)^{\frac{3}{2}}+C\\ &=-\frac{1}{3a^2}\left(\frac{a^2}{x^2}-1\right)^{\frac{3}{2}}+C=-\frac{1}{3a^2x^3}\left(a^2-x^2\right)^{\frac{3}{2}}+C\end{align*}

3，万能代换：也称为半角代换，就是做代换 $$\displaystyle\quad x=\tan \frac{t}{2}$$，$$\displaystyle\quad t=\tan \frac{x}{2}$$，这样的话，

$\sin x=\frac{2\tan \frac{x}{2}}{1+\tan \frac{x}{2}}=\frac{2t}{1+t^2},\cos x=\frac{1-\tan^2 \frac{x}{2}}{1+\tan^2 \frac{x}{2}}=\frac{1-t^2}{1+t^2}$