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Table of Integrals

1. \(\int dx=x+C\) 2. \(\int x^{\alpha}dx=\frac{x^{\alpha+1}}{\alpha+1}+C\) 3. \(\int \frac{dx}{x}=\ln |x|+C\) 4. \(\int a^x dx=\frac{a^x}{\ln a}+C\) 5. \(\int e^x dx=e^x+C\) 6. \(\int \sin x dx=-\cos x+C\) 7. \(\int \cos x dx=\sin x+C\) 8. \(\int \frac{dx}{\cos^2 x}=tg x+C\) 9. \(\int \frac{dx}{sin^2 x}=-ctg x+C\) 10. \(\int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin\frac{x}{a}+C\) 11. \(\int \frac{dx}{\sqrt{x^2 \pm a^2}}=\ln\left|x+\sqrt{x^2\pm a^2}\right|+C\) 12. \(\int \frac{dx}{x^2+a^2}=\frac{1}{a}arctg\frac{x}{a}+C\) 13. \(\int […]

Table of Higher Order Derivatives

\((x^p)^{(n)}= p(p-1)(p-2)…(p-n+1)x^{p-n}.\) \( (a^x)^{(n)}=a^x\ln^na \qquad\qquad(e^x)^{(n)}=e^x\) \((\sin \alpha x)^{(n)}=\alpha^n\sin\left(\alpha x+\frac{\pi n}{2}\right)\) \((\cos \alpha x)^{(n)}=\alpha^n\cos\left(\alpha x+\frac{\pi n}{2}\right)\) \(\left((ax+b)^p\right)^{(n)}=a^np(p-1)(p-2)…(p-n+1)(ax+b)^{p-n}\) \((\log_a |x|)^{(n)}=\frac{(-1)^{n-1}(n-1)!}{x^n\ln a}\) \((\ln |x|)^{(n)}=\frac{(-1)^{n-1}(n-1)!}{x^n}\) \((\alpha u(x)+\beta v(x))^{(n)}=\alpha u^{(n)}(x)+\beta v^{(n)}(x)\) \((u(x)v(x))^{(n)}=\sum\limits_{k=0}^n C_n^k u^{(k)}(x)v^{(n-k)}(x),\,\, \mbox {where}\quad C_n^k=\frac{n!}{k!(n-k)!}\)

Table of Derivatives Complex Functions

\(c’=0,\quad  c=const;\) \((u(x)^\alpha)’=\alpha u^{\alpha-1}(x)u’(x), \quad x\in \mathbb{R}, \alpha\in \mathbb{R};\) \((a^{u(x)})’=a^{u(x)}\ln a\cdot u’(x),\quad a>0, a\neq 1, x\in \mathbb{R};\) \((e^u(x))’=e^{u(x)}u’(x);\) \((\log_a u(x))’=\frac{1}{u(x)\ln a}u’(x), \quad x>0;\) \((\log_a|u(x)|)’=\frac{1}{u(x)\ln a}u’(x),\quad x\neq 0;\) \((\ln u(x))’=\frac{1}{u(x)}u’(x),\quad x>0;\) \((\sin u(x))’=\cos u(x) \cdot u’(x), \quad x\in \mathbb{R};\) \((\cos u(x))’=-\sin u(x) \cdot u’(x)\quad x\in \mathbb{R};\) \((\mathrm{tg} u’(x))’=\frac{1}{\cos^2 u’(x)}u’(x),\quad x\neq \frac{\pi}{2}(2n+1), \in \mathbb{Z};\) \((\mathrm{ctg} u(x))’=-\frac{1}{\sin^2 u(x)}u’(x),\quad […]

Table of Derivatives Formulas

\(c’=0,\quad  c=const;\) \((x^\alpha)’=\alpha x^{\alpha-1}, \quad x\in \mathbb{R}, \alpha\in \mathbb{R};\) \((a^x)’=a^x\ln a,\quad a>0, a\neq 1, x\in \mathbb{R};\) \((e^x)’=e^x;\) \((\log_a x)’=\frac{1}{x\ln a}, \quad x>0;\) \((\log_a|x|)’=\frac{1}{x\ln a},\quad x\neq 0;\) \((\ln x)’=\frac{1}{x},\quad x>0;\) \((\sin x)’=\cos x, \quad x\in \mathbb{R};\) \((\cos x)’=-\sin x\quad x\in \mathbb{R};\) \((\mathrm{tg} x)’=\frac{1}{\cos^2 x},\quad x\neq \frac{\pi}{2}(2n+1), n\in \mathbb{Z};\) \((\mathrm{ctg} x)’=-\frac{1}{\sin^2 x},\quad x\neq \pi n, n\in \mathbb{Z};\) […]

Taylor and Maclaurin Series

Definition of Taylor series: \( f(x) = f(a) + f’(a)(x-a) + \frac{f”(a)(x-a)^2}{2!} + \cdots +\frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} + R_n \) \( R_n = \frac{f^{(n)}(\xi)(x-a)^n}{n!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Lagrangue’s form )} \) \( R_n = \frac{f^{(n)}(\xi)(x-\xi)^{n-1}(x-a)}{(n-1)!} \text{ where } a \leq \xi \leq x, \quad \text{ ( Cauch’s form )} \) This result […]

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