# 趋同和去分母是积分运算中常用的解题思路

## 二、解析

### 解法一：趋同

\begin{aligned} I \\ \\ & = \int \frac{x \textcolor{orangered}{\tan x} }{\cos ^{ 4 } x} \mathrm{~d} x \\ \\ & = \int \frac{x}{\cos ^{ 4 } x} \cdot \textcolor{orangered}{ \frac{\sin x}{\cos x} } \mathrm{~d} x \\ \\ & = \int \frac { x } { \cos ^ { 5 } x } \cdot \textcolor{springgreen}{ \sin x \mathrm { ~ d } x } \\ \\ & = – \int \frac { x } { \cos ^ { 5 } x } \textcolor{springgreen}{ \mathrm { ~ d } \cos x } \end{aligned}

$$\textcolor{yellow}{ \left( \frac{1}{\cos ^{ 4 } x} \right) ^{\prime} _{\cos x} = \frac{- 4 \cos ^{ 3 } x}{\cos ^{ 8 } x} = -4 \cdot \frac{1}{\cos ^{ 5 } x} }$$

\begin{aligned} I \\ \\ & = \frac { 1 } { 4 } \int x \mathrm { ~ d } \left( \frac { 1 } { \cos ^ { 4 } x } \right) \\ \\ & = \frac { x } { 4 \cos ^ { 4 } x } – \frac { 1 } { 4 } \int \frac { 1 } { \cos ^ { 4 } x } \mathrm { ~ d } x \end{aligned}

\textcolor{yellow}{ \begin{aligned} \frac{1}{\cos ^{4} x} \\ \\ & = \frac{1}{\cos ^{2} x} \cdot \frac{1}{\cos ^{2} x} \\ \\ & = \frac{1}{\cos ^{2} x} \mathrm{~d} (\tan x) \\ \\ & = \frac{\sin ^{2} x + \cos ^{2} x}{\cos ^{2} x} \mathrm{~d} (\tan x) \\ \\ & = \left( \tan ^{2} x + 1 \right) \mathrm{~d} ( \tan x) \end{aligned} }

\begin{aligned} I \\ \\ & = \frac { x } { 4 \cos ^ { 4 } x } – \frac { 1 } { 4 } \int \left( \tan ^ { 2 } x + 1 \right) \mathrm { d } \tan x \\ \\ & = \frac { x } { 4 \cos ^ { 4 } x } – \frac { 1 } { 1 2 } \tan ^ { 3 } x – \frac { 1 } { 4 } \tan x + C \end{aligned}

### 解法二：去分母

\begin{aligned} I \\ \\ & = \int \frac { \textcolor{yellow}{ x \tan x } } { \cos ^ { 4 } x } \mathrm { ~ d } x \\ \\ & = \int \textcolor{yellow}{ x \tan x } \cdot \textcolor{springgreen}{\frac { 1 } { \cos ^ { 4 } x } \mathrm { ~ d } x } \\ \\ & = \int \textcolor{yellow}{ x \tan x } \cdot \textcolor{springgreen}{ \sec ^ { 2 } x \sec ^ { 2 } x \mathrm { ~ d } x } \end{aligned}

\begin{aligned} I \\ \\ & = \int \textcolor{yellow}{ x \tan x } \cdot \textcolor{springgreen}{ \left( 1 + \tan ^ { 2 } x \right) \mathrm { d } \tan x } \\ \\ & = \int x \textcolor{red}{ \tan x \cdot \left( 1 + \tan ^ { 2 } x \right) \mathrm { d } \tan x } \end{aligned}

$$\textcolor{yellow}{ \left[ \frac{1}{4} (1+ t ^{2}) ^{2} \right] ^{\prime} = t \left( 1 + t ^{2} \right) }$$

\begin{aligned} I \\ \\ & = \int x \textcolor{red}{ \mathrm { ~ d } \left( \frac { 1 } { 4 } \left( 1 + \tan ^ { 2 } x \right) ^ { 2 } \right) } \\ \\ & = \frac { 1 } { 4 } x \left( 1 + \tan ^ { 2 } x \right) ^ { 2 } – \textcolor{springgreen}{ \frac { 1 } { 4 } \int \left( 1 + \tan ^ { 2 } x \right) ^ { 2 } \mathrm { ~ d } x } \end{aligned}

$$\textcolor{yellow}{ 1 + \tan ^{2} x = \frac{1}{\cos ^{2} x} = (\tan x) ^{\prime} _{x} }$$

\begin{aligned} I \\ \\ & = \frac { 1 } { 4 } x \left( 1 + \tan ^ { 2 } x \right) ^ { 2 } – \textcolor{springgreen}{ \frac { 1 } { 4 } \int \left( 1 + \tan ^ { 2 } x \right) \left( 1 + \tan ^ { 2 } x \right) \mathrm { ~ d } x } \\ \\ & = \frac { 1 } { 4 } x \left( 1 + \tan ^ { 2 } x \right) ^ { 2 } – \textcolor{springgreen}{ \frac { 1 } { 4 } \int \left( 1 + \tan ^ { 2 } x \right) \mathrm { d } \tan x } \\ \\ & = \frac { 1 } { 4 } x \left( 1 + \tan ^ { 2 } x \right) ^ { 2 } – \textcolor{springgreen}{\frac { 1 } { 4 } \tan x – \frac { 1 } { 1 2 } \tan ^ { 3 } x } + C \\ \\ & = \frac { x } { 4 \cos ^ { 4 } x } – \frac { 1 } { 4 } \tan x – \frac { 1 } { 1 2 } \tan ^ { 3 } x + C \end{aligned}