$$ \small\int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin{\frac xa}+C\\ \int \frac{dx}{a^2+x^2}=\frac 1a \arctan \frac xa + C\\ \int \frac{dx}{x\sqrt{x^2-a^2}}=\frac 1a \text{arcsec}\left|\frac xa\right|+C $$

• 중간

7.4

Newton’s Law of Cooling $\frac{dH}{dt}=-k(H-H_s)$

7.5

L’Hopital’s Rule $\displaystyle\lim_{x\to a}f(x)=\lim_{g\to a}g(x)=0$이고 열린 구간 $I(\ni a)$에서 미분 가능하며 $g'(x)\neq0\ (x\neq a)$이면

$\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}.$ 추가로 $x\to\pm\infty$ 또는 $f', g'\to \pm \infty$ 에서도 성립.

• Cauchy’s Mean Value Theorem: f, g가 [a, b]에서 연속이고 (a, b)에서 미분가능 & g’≠0일 때 $\exists c\in (a, b) \text{ s.t. } \frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.$

7.6, 7.7

$\frac{d}{dx}(\sin^{-1} x)=\cfrac{1}{\sqrt{1-x^2}},\,\frac{d}{dx}(\tan^{-1} x)=\cfrac{1}{1+x^2},\,\frac{d}{dx}(\sec^{-1}x)=\cfrac{1}{|x|\sqrt{x^2-1}}$

• $\small\arccos x = \pi/2-\arcsin x,\,\ldots$

$\sinh x=\cfrac{e^x-e^{-x}}2,\, \cosh x=\cfrac{e^x+e^{-x}}2$

Numerical Integration

Midpoint Rule: $\int_a^b f(x)\,\mathrm{d}x\approx\sum^n_{k=1}f\left(\frac{x_{k-1}+x_k}{2}\right)\left(\frac{b-a}{n}\right)$, $|E_M|\le \dfrac{M(b-a)^3}{24n^2}$

Trapezoidal Rule: $\frac{\Delta x}{2}(y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + 2y_n)$, $|E_T|\le\dfrac{M(b-a)^3}{12n^2}$

Simpson’s Rule: $\frac{\Delta x}{3}(y_0 + 4y_1 + 2y_2 + 4y_3 + \cdots + 4y_{n-1} + y_n)$, $|E_S|\le \dfrac{M(b-a)^5}{180n^4}$

$M$은 $|f''|, |f''|, |f^{(4)}|$ 의 최댓값