###### tags: `課業`
# Santa calculus Answers
## Speedrun
* $\frac{d}{dx}(\cos x)=-\sin x$
* $\int \sin (2x) dx=-\frac{1}{2} \cos 2x +C$
* $\frac{d}{dx}(\tan x) =\sec^2 x$
* $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$
* $\int \cot x \csc x dx = -\csc x + C$
* $\int \sec x dx = \ln{|\sec x + \tan x|}+C$

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* $\frac{d}{dx}(\cot (x^2))=- \csc ^2 x^2 \cdot 2x$
* $\int \tan x dx=-\ln {|\cos x|}+C$
* $\frac{d}{dx}(\ln x)=\frac{1}{x}$
* $\int_{-2}^{2} \frac{\sin (x^3)}{x^8+1} dx = 0$
* $\int_1^2 x\lfloor 2x \rfloor dx = 3.875$
* $\frac{d}{dx}(\frac{1}{x}) = \frac{-1}{x^2}$

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* $\int \frac{1}{x} dx = \ln {|x|}+C$
* $\int \sin x \cos x dx = -\frac{1}{4} \cos 2x + C$
* $\int_0^2 \sqrt{4-x^2} dx = \pi$
* $\int \ln x dx=x \ln x -x+C$
* $\frac{d}{dx}({5}^{x^2})=\ln 25 \cdot {5}^{x^2} \cdot x$
* If $f(x)=x^7+x^5+x^3+1$, then $(f^{-1})'(4)=\frac{1}{15}$.

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* $\lim\limits_{x \rightarrow 0} x \sin (\frac{1}{x})=0$
* $\lim\limits_{x \rightarrow \infty} x \sin (\frac{1}{x})=1$
* $\lim\limits_{x \rightarrow 0} \frac{x}{\tan x}=1$
* $\lim\limits_{x \rightarrow \infty} \frac{\lfloor x \rfloor}{x}=1$
* $\lim\limits_{x \rightarrow \infty} x^\frac{1}{\ln x}=e$
* $\lim\limits_{x \rightarrow \infty} (1-\frac{2}{x})^{x}=\frac{1}{e^2}$

## 單選
1. A
2. A
3. B
4. C
5. C
6. D
7. C
8. D
9. D
10. B

## 多選
11. B
12. BC
13. AD
14. BD
15. A

## 計算證明

1. (A) $I_n=\frac{-1}{n} \cos x \sin^{n-1} x + \frac{n-1}{n} I_{n-2}$
(B) \begin{equation} \begin{cases} \frac{(n-1)(n-3)...2}{n(n-2)...1},\quad n \space \mathrm{is} \space \mathrm{odd} \\ \frac{(n-1)(n-3)...1}{n(n-2)...2}\frac{\pi}{2}, \quad n \space \mathrm{is} \space \mathrm{even}\end{cases} \end{equation}

2. (A) $1 + \ln{\frac 5 3}$
(B) $4\pi(2\sqrt{5} - \sqrt{2} + \ln{\frac{2+\sqrt 5}{1 + \sqrt 2}})$