Important Theorems in Calculus

Important Theorems in Calculus

Calculus allows us to get analyses of a function’s behavior over the time. Calculus includes the derivatives and integrals. The basic theorem in calculus defines the relationship between derivatives and integrals. There are several theorems that define important features of the derivative and integral values of functions. Fundamental Theorem of Calculus The first part of […]

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 […]

Functions to Know in Calculus

Calculus is part of mathematics that deals with change. In algebra, you learned how to graph polynomials while in calculus you will learn how the graphed curve of polynomials change direction at each point. Limits Limits are the first topic when we study the calculus. Limit is the sequence 1/2, 1/3, 1/4 etc. It is […]

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 […]

Page 3 of 1912345...10...»»