# 凑微分：一道积分题用凑微分能有多少种解法？

$$I=\int \frac{\mathrm{d} x}{\sin x \cos x} = ?$$

\begin{aligned} I = & \int \frac{1}{\sin x \cos x} \mathrm{~d} x \\ = & \int \frac{\sin ^{2} x+\cos ^{2} x}{\sin x \cos x} \mathrm{~d} x \\ = & \int \frac{\sin ^{2} x}{\sin x \cos x} \mathrm{~d} x+\int \frac{\cos ^{2} x}{\sin x \cos x} \mathrm{~d} x \\ = & \int \frac{\sin x}{\cos x} \mathrm{~d} x+\int \frac{\cos x}{\sin x} \mathrm{~d} x \\ = & -\ln |\cos x|+\ln |\sin x| + C \\ = & \ln \left|\frac{\sin x}{\cos x}\right| + C = \ln |\tan x| + C \end{aligned}

\begin{aligned} I = & \int \frac{1}{\sin x \cos x} \mathrm{~d} x \\ = & \int \frac{1 / \cos ^{2} x}{\sin x / \cos x} \mathrm{~d} x \\ = & \int \frac{1}{\tan x} \mathrm{~d} (\tan x)=\ln |\tan x|+C \end{aligned}

\begin{aligned} I = & \int \frac{1}{\sin x \cos x} \mathrm{~d} x \\ = & \int \frac{\cos x}{\sin x \cos ^{2} x} \mathrm{~d} x \\ = & \int \frac{\mathrm{~d} (\sin x)}{\sin x\left(1-\sin ^{2} x\right)} \mathrm{~d} x \Rightarrow \textcolor{springgreen}{t=\sin x} \Rightarrow \\ = & \int \frac{1}{t\left(1-t^{2}\right)} \mathrm{~d} t \end{aligned}

$$I = \frac{A}{t}+\frac{B t+C}{1-t^{2}} \Rightarrow$$

$$I = \frac{A-A t^{2}+B t^{2}+C t}{t\left(1-t^{2}\right)} \Rightarrow$$

$$\begin{cases} & A=1 \\ & B=1 \\ & C=0 \end{cases}$$

\begin{aligned} I = & \int\left(\frac{1}{t}+\frac{t}{1-t^{2}}\right) \mathrm{~d} t \\ = & \ln |t|-\frac{1}{2} \ln \left|1-t^{2}\right| + C \\ = & \ln \left|\frac{t}{\sqrt{1-t^{2}}}\right| + C \\ = & \ln \left|\frac{\sin x}{\cos x}\right| + C = \ln |\tan x|+C \end{aligned}

\begin{aligned} I = & \int \frac{1}{\sin x \cos x} \mathrm{~d} x \\ = & \int \frac{1}{1 / 2 \sin 2 x} \mathrm{~d} x= \\ = & 2 \cdot \frac{1}{2} \int \frac{1}{\sin 2 x} \mathrm{~d} (2 x) \\ = & \int \frac{1}{\sin 2 x} \mathrm{~d} (2 x) \end{aligned}

\begin{aligned} (\ln |\tan x|)^{\prime} = & \frac{\tan x}{\frac{1}{\cos ^{2} x}} \\ = & \frac{\cos x}{\sin x} \cdot \frac{1}{\cos ^{2} x} \\ = & \frac{2}{\sin x \cos x} \end{aligned}

\begin{aligned} I = & \int \frac{1}{\sin 2 x} \cdot \mathrm{~d} (2 x) \\ = & \ln \left|\tan \frac{2 x}{2}\right|+C = \ln |\tan x|+C \end{aligned}