###### tags: `微積分筆記`

# 積分筆記 $\begin{align}\end{align}$

## 第一類 : 根式
1. $\begin{align}\int x\sqrt{a-x^2}dx=-\frac{1}{3}(a-x^2)^{\frac{3}{2}}+C\end{align}$

2. $\begin{align}\int \sqrt{a-x^2}dx\end{align}$
$\begin{align}令x=\sqrt{a}\sin \theta,dx=\sqrt{a}\cos \theta d\theta\end{align}$
$\begin{align}原式=\int\sqrt{a}\cos \theta\times(\sqrt{a}\cos \theta)d\theta=\int a\cos^2 \theta d\theta\end{align}$
$\begin{align}=a\int \frac{1+\cos 2\theta}{2} d\theta=\frac{a}{2}\theta-\frac{a}{4}\sin 2\theta=\frac{a}{2}\sin^{-1}\frac{x}{\sqrt{a}}-\frac{a}{4}\cdot2\cdot\frac{x}{\sqrt{a}}\cdot\frac{\sqrt{x^2-a}}{x}\end{align}$
$\begin{align}=\frac{a}{2}\sin^{-1}\frac{x}{\sqrt{a}}-\frac{\sqrt{a}\cdot\sqrt{x^2-a}}{2}\end{align}$
3. $\begin{align}\int x^2\sqrt{a-x^2}dx\end{align}$
$\begin{align}令x=\sqrt{a}\sin \theta,dx=\sqrt{a}\cos \theta d\theta\end{align}$
$\begin{align}原式=\int a\sin^2 \theta \cdot a\cdot\cos^2 \theta d\theta=\int a^2\sin^2 \theta(1-\sin^2 \theta)d\theta=a^2\int \sin^2 \theta d\theta-a^2\int\sin^4\theta d\theta\end{align}$
$\begin{align}=a^2\int \frac{1}{2}-\frac{\cos 2\theta}{2}d\theta-\end{align}$
$\begin{align}=a^2(\frac{1}{2}\sin^{-1}\frac{x}{\sqrt{a}}+\frac{x^2-a}{2\sqrt{a}})-\end{align}$
4. $\begin{align}\int x^b\sqrt{a-x^2}dx\end{align}$

## 第二類：分式多項式

1. $\begin{align}\int\frac{1}{ax+b} dx=\frac{1}{a}\ln|ax+b|+c\end{align}$

2. $\begin{align}\int\frac{1}{(x+2)(x+3)} dx=\int\frac{1}{x+2}-\frac{1}{x+3} dx\end{align}$

3. $\begin{align}\int\frac{1}{x^2+x+1}dx=\int\frac{1}{(x+\frac{1}{2})^2+\frac{3}{4}}dx\end{align}$
令 $\begin{align}x+\frac{1}{2}=\frac{\sqrt{3}}{2}\tan\theta，dx=\frac{\sqrt{3}}{2}\sec^2\theta d\theta\end{align}$
4. $\begin{align}\int\frac{2x+3}{x^2+x+1}dx=\int\frac{2x+1+2}{x^2+x+1}dx\end{align}$
$\begin{align}=\ln(x^2+x+1)+\int\frac{2}{x^2+x+1} dx\end{align}$
5.

## 第三類 : 分部積分

1. $\begin{align}\int x\sin xdx=x\cdot(-\cos x)-\int(-\cos x)dx=-x\cos x+\sin x+c\end{align}$

2. $\begin{align}\int e^x\sin xdx=e^x(-\cos x)-e^x(-\sin x)+\int e^x(-\sin x)dx\end{align}$
$\begin{align}\int e^x\sin xdx=\frac{1}{2}e^x(\sin x -\cos x)+c\end{align}$
3. $\begin{align}\int \sec^3 xdx\end{align}$
$\begin{align}u=\sec x,v'=\sec^2 x\end{align}$
$\begin{align}\int\sec^3 x=\sec x\tan x-\int\sec x\tan^2 xdx=\sec x\tan x-\int(\sec^3 x-\sec x)dx\end{align}$
$\begin{align}\int\sec^3 xdx=\frac{1}{2}(\sec x\tan x-\int\sec xdx)=\frac{1}{2}(\sec x\tan x-\ln(\sec x+\tan x))+c\end{align}$
4. $\begin{align}\int \ln x dx=\int\ln\cdot1dx=x\ln x-\int x\cdot\frac{1}{x}dx=x(\ln x-1) \end{align}$

5. $\begin{align}\int x\ln x dx=\frac{x^2}{2} \ln x-\int\frac{x^2}{2}\cdot\frac{1}{x}dx=\frac{1}{2}x^2\ln x-\frac{1}{4}x^2+c\end{align}$

6. $\begin{align}\int \end{align}$

## 第三類 :