推薦的微積分線上計算機 https://zs.symbolab.com/solver/pre-calculus-calculator

微分常用公式

$$(f(x) + g(x))' = f'(x) + g'(x)$$
$$(f(x) - g(x))' = f'(x) - g'(x)$$
$$(cf(x))' = c f'(x) \quad \text{（c為常數）}$$
$$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \quad \text{（乘法法則）}$$
$$\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \quad \text{（除法法則）}$$

$$(f(g(x)))' = f'(g(x)) \cdot g'(x) \quad \text{（鏈式法則）}$$

$$\frac{d}{dx}(x^n) = nx^{n-1}$$
$$\frac{d}{dx}(\sin x) = \cos x$$
$$\frac{d}{dx}(\cos x) = -\sin x$$
$$\frac{d}{dx}(\tan x) = \sec^2 x$$
$$\frac{d}{dx}(e^x) = e^x$$
$$\frac{d}{dx}(\ln x) = \frac{1}{x}$$

積分常用公式

https://youtu.be/yvr4kjsqFFM?list=PLP1Ynr8cs97utEQe5VP3cv_iscJVuwbog

反導數(Anti Derivatives)

定義：若F'(x) = f(x)

稱F(x)為f(x)的反(導數)微分

$$\int f(x) dx = F(x) + C$$

一個函數先積分在微分會等同於自己

$$ \frac{d}{dx} \int f(x) dx = f(x)$$

先微分在積分也是一樣的道理，會等同於自己

$$ \int f'(x) dx = f(x)+C$$

例題

$$ \int x^3 dx = \frac{1}{4}\ x^4+C $$

左邊$x^3$，是右邊多少微分過去的呢？想要微出$x^3$，那右邊想必就是 $$\frac{1}{4}\ x^4+C $$ 才會有這個答案

例題

$$ \int e^x dx = e^x+C $$

例題

$$ \int cos x dx = sinx+C $$

基本性質

https://youtu.be/8Ccx-dYWm9k?list=PLP1Ynr8cs97utEQe5VP3cv_iscJVuwbog

$$ \int k dx = kx+C $$

$$ \int kf(x) dx = k \int f(x) dx $$

$$ ∫[f(x)±g(x)]dx=∫f(x)dx±∫g(x)dx $$

$$ \int x^n dx = k \int f(x) dx $$

$$
\int x^n , dx = \begin{cases} \frac{1}{n+1} x^{n+1} + C & \text{if } n \ne -1 \
\ln |x| + C & \text{if } n = -1 \end{cases} \ $$

變數變換法

把多項式先用某個變數替代，之後解出答案後再還原，如下

進階變數變換法

定積分公式

$$ \begin{aligned} &\text{計算 } \int_1^3 (2x + 1),dx \\
&\textbf{步驟 1：} \text{先對 } 2x + 1 \text{ 做不定積分} \\
&\int (2x + 1),dx = x^2 + x + C \\
&\textbf{步驟 2：} \text{代入上下限} \\
&\left[ x^2 + x \right]_1^3 = (3^2 + 3) - (1^2 + 1) \\
&= (9 + 3) - (1 + 1) = 12 - 2 = 10 \\
&\therefore \int_1^3 (2x + 1),dx = \boxed{10} \end{aligned} $$