最近在清理电脑，发现有些题目之前写的，没有保存，这里记录一下

求行列式

$$ D = \begin{vmatrix} \sin(\theta_1+\theta_1)&\sin(\theta_1+\theta_2)&\cdots&\sin(\theta_1+\theta_n)\\
\sin(\theta_2+\theta_1)&\sin(\theta_2+\theta_2)&\cdots&\sin(\theta_2+\theta_n)\\
\vdots&\vdots&&\vdots\\
\sin(\theta_n+\theta_1)&\sin(\theta_n+\theta_2)&\cdots&\sin(\theta_n+\theta_n) \end{vmatrix} $$

【Sol】: 当 $n\geqslant2$ 时

$$ \begin{aligned} &\begin{vmatrix} \sin(\theta_1+\theta_1)&\sin(\theta_1+\theta_2)&\cdots&\sin(\theta_1+\theta_n)\\
\sin(\theta_2+\theta_1)&\sin(\theta_2+\theta_2)&\cdots&\sin(\theta_2+\theta_n)\\
\vdots&\vdots&&\vdots\\
\sin(\theta_n+\theta_1)&\sin(\theta_n+\theta_2)&\cdots&\sin(\theta_n+\theta_n) \end{vmatrix}\\
=&\begin{vmatrix} \sin\theta_1&\cos\theta_1&0&\cdots&0\\
\sin\theta_2&\cos\theta_2&0&\cdots&0\\
\vdots&\vdots&\vdots&&\vdots\\
\sin\theta_n&\cos\theta_n&0&\cdots&0 \end{vmatrix}_ {n \times n} \cdot \begin{vmatrix} \cos\theta_1&\cos\theta_2&\cdots&\cos\theta_n\\
\sin\theta_1&\sin\theta_2&\cdots&\sin\theta_n\\
0&0&\cdots&0\\
\vdots&\vdots&&\vdots\\
0&0&\cdots&0 \end{vmatrix}_{n\times n}\\
=&\begin{cases} -\sin^2(\theta_1-\theta_2),&n=2\\
0,&n\geqslant3 \end{cases} \end{aligned} $$

因此

$$ D= \begin{cases} \sin2\theta_1,&n=1\\
-\sin^2(\theta_1-\theta_2),&n=2\\
0,&n\geqslant3 \end{cases} $$

求行列式

$$ \begin{vmatrix} a_1^{n-1}&a_1^{n-2}b_1&\cdots&a_1b_1^{n-2}&b_1^{n-1}\\
a_2^{n-1}&a_2^{n-2}b_2&\cdots&a_2b_2^{n-2}&b_2^{n-1}\\
\vdots&\vdots&&\vdots&\vdots\\
a_{n-1}^{n-1}&a_{n-1}^{n-2}b_{n-1}&\cdots&a_{n-1}b_{n-1}^{n-2}&b_{n-1}^{n-1}\\
a_{n}^{n-1}&a_{n}^{n-2}b_n&\cdots&a_{n}b_n^{n-2}&b_{n}^{n-1} \end{vmatrix} $$

【Sol】:
首先引入引理，范德蒙(Vandermonde)行列式:

$$ D_n=\begin{vmatrix} 1&1&\cdots&1\\
x_1&x_2&\cdots&x_n\\
\cdots&\cdots&&\cdots\\
x_1^{n-2}&x_2^{n-2}&\cdots&x_{n}^{n-2}\\
x_1^{n-1}&x_{2}^{n-1}&\cdots&x_{n}^{n-1} \end{vmatrix} = \prod_{1\leqslant i<j\leqslant n}(x_j-x_i) $$

$$ \begin{aligned} &\begin{vmatrix} a_1^{n-1}&a_1^{n-2}b_1&\cdots&a_1b_1^{n-2}&b_1^{n-1}\\
a_2^{n-1}&a_2^{n-2}b_2&\cdots&a_2b_2^{n-2}&b_2^{n-1}\\
\vdots&\vdots&&\vdots&\vdots\\
a_{n-1}^{n-1}&a_{n-1}^{n-2}b_{n-1}&\cdots&a_{n-1}b_{n-1}^{n-2}&b_{n-1}^{n-1}\\
a_{n}^{n-1}&a_{n}^{n-2}b_n&\cdots&a_{n}b_n^{n-2}&b_{n}^{n-1} \end{vmatrix}\\
=&\prod_{i=1}^{n-1}a_i^{n-1} \begin{vmatrix} 1&\dfrac{b_1}{a_1}&\cdots&\left(\dfrac{b_1}{a_1}\right)^{n-2}&\left(\dfrac{b_1}{a_1}\right)^{n-1}\\
1&\dfrac{b_2}{a_2}&\cdots&\left(\dfrac{b_2}{a_2}\right)^{n-2}&\left(\dfrac{b_2}{a_2}\right)^{n-1}\\
\vdots&\vdots&&\vdots&\vdots\\
1&\dfrac{b_{n-1}}{a_{n-1}}&\cdots&\left(\dfrac{b_{n-1}}{a_{n-1}}\right)^{n-2}&\left(\dfrac{b_{n-1}}{a_{n-1}}\right)^{n-1}\\
1&\dfrac{b_n}{a_n}&\cdots&\left(\dfrac{b_n}{a_n}\right)^{n-2}&\left(\dfrac{b_n}{a_n}\right)^{n-1} \end{vmatrix}\\
=&\prod_{i=1}^{n-1}a_i^{n-1}\prod_{1\leqslant i<j\leqslant n}(\dfrac{b_j}{a_j}-\dfrac{b_i}{a_i}) \end{aligned} $$

几个结论

$\displaystyle\int_0^{+\infty}e^{-x}x^5\text{d}x=120$

$\displaystyle\int_{-\infty}^{+\infty}e^{-\frac{x^2}{2}}\text{d}=\sqrt{2\pi}$

$\displaystyle\int_0^{+\infty}\sin x^2\text{d}x=\dfrac{\sqrt{\pi}}{2\sqrt{2}}$

$\displaystyle\int_0^{+\infty}\dfrac{\sin x}{x}\text{d}x=\dfrac{\pi}{2}$

$\displaystyle\sum_{n=0}^{\infty}\dfrac{(-1)^n}{2n+1}=\dfrac{\pi}{4}$

$\displaystyle\arctan\dfrac{2}{n^2}=\arctan(n+1)-\arctan(n-1)$

$\displaystyle\sum_{n=1}^{\infty}\arctan\dfrac{2}{n^2}=\lim_{n\to\infty}\arctan(n+1)+\arctan n-\arctan1=\dfrac{3\pi}{4}$

$\displaystyle\lim_{t\to0^+}\int_{-2020}^{2020}\dfrac{t\cos x}{x^2+t^2}\text{d}x=\lim_{t\to0^+}\dfrac{\pi e^{-t}}{t}\cdot t=\pi$