Fundamental Theorem of Calculus (FTC)

Fundamental Theorem of Calculus (FTC)Calculus I RulesCC BY-NC-SA 4.0

Author: John J Weber III, PhD Corresponding Textbook Sections:

Section 4.9 – Antiderivatives
Section 5.3 – The Fundamental Theorem of Calculus
Section 5.4 – Indefinite Integrals and the Net Change Theorem

Calculus I Rules

NOTE: You are not permitted to use this table on any Assessment.

Function	Derivative	Function	Antiderivative
$\displaystyle f(x)$	$\displaystyle f^{\,\prime}(x)$	$\displaystyle f(x)$	$\displaystyle F(x)$
$\displaystyle f(x)=a$ $a\in\mathbb{R}$	$\displaystyle f^{\,\prime}(x)=0$	$\displaystyle f(x)=a$ $a\in\mathbb{R}$	$\displaystyle F(x)=ax+C$
$\displaystyle f(x)=ax^n$ $a,n\in\mathbb{R}$	$\displaystyle f^{\,\prime}(x)=nax^{n-1}$	$\displaystyle f(x)=ax^n$ $a,n\in\mathbb{R}$ $n\ne -1$	$\displaystyle F(x)=\frac{x^{n+1}}{n+1}+C$
$\displaystyle f(x)=\ln{(x)}$	$\displaystyle f^{\,\prime}(x)=\frac{1}{x}$	$\displaystyle f(x)=\frac{1}{x}=x^{-1}$	$\displaystyle F(x)=\ln{(x)}+C$
$\displaystyle f(x)=\log_b{(x)}$ $b>0,b\ne 1$ $x>0$	$\displaystyle f^{\,\prime}(x)=\frac{1}{x\,\ln{(b)}}$	$\displaystyle f(x)=\frac{1}{x\,\ln{(b)}}$ $b>0,b\ne 1$	$\displaystyle F(x)\!=\!\left(\!\frac{1}{\ln{\!(b)}}\!\right)\!\ln{\!(x)}\!+\!C$
$\displaystyle f(x)=a^x$ $a>0,a\ne 1$	$\displaystyle f^{\,\prime}(x)=\ln{(a)}\,a^x$	$\displaystyle f(x)=a^x$ $a>0,a\ne 1$	$\displaystyle F(x)=\frac{a^x}{\ln{(a)}}+C$
$\displaystyle f(x)=e^x$	$\displaystyle f^{\,\prime}(x)=e^x$	$\displaystyle f(x)=e^x$	$\displaystyle F(x)=e^x+C$
$\displaystyle f(x)=\sin{(x)}$	$\displaystyle f^{\,\prime}(x)=\cos{(x)}$	$\displaystyle f(x)=\cos{(x)}$	$\displaystyle F(x)=\sin{(x)}+C$
$\displaystyle f(x)=\cos{(x)}$	$\displaystyle f^{\,\prime}(x)=-\sin{(x)}$	$\displaystyle f(x)=\sin{(x)}$	$\displaystyle F(x)=-\cos{(x)}+C$
$\displaystyle f(x)=\tan{(x)}$	$\displaystyle f^{\,\prime}(x)=\sec^2{(x)}$	$\displaystyle f(x)=\sec^2{(x)}$	$\displaystyle F(x)=\tan{(x)}+C$
$\displaystyle f(x)=\cot{(x)}$	$\displaystyle f^{\,\prime}(x)=-\csc^2{(x)}$	$\displaystyle f(x)=\csc^2{(x)}$	$\displaystyle F(x)=-\cot{(x)}+C$
$\displaystyle f(x)=\sec{(x)}$	$\displaystyle f^{\,\prime}(x)\!=\!\sec{(x)}\tan{(x)}$	$\displaystyle f(x)\!=\!\sec{\!(x)}\tan{\!(x)}$	$\displaystyle F(x)=\sec{(x)}+C$
$\displaystyle f(x)=\csc{(x)}$	$\displaystyle f^{\,\prime}(x)\!=\!-\csc{\!(x)}\cot{\!(x)}$	$\displaystyle f(x)\!=\!\csc{\!(x)}\cot{\!(x)}$	$\displaystyle F(x)=-\csc{(x)}+C$
$\displaystyle f(x)=\sin^{-1}\!{(x)}$	$\displaystyle f^{\,\prime}(x)=\frac{1}{\sqrt{1-x^2}}$	$\displaystyle f(x)=\frac{1}{\sqrt{1-x^2}}$	$\displaystyle F(x)=\sin^{-1}{(x)}+C$
$\displaystyle f(x)=\cos^{-1}\!{(x)}$	$\displaystyle f^{\,\prime}(x)=-\frac{1}{\sqrt{1-x^2}}$	$\displaystyle f(x)=\frac{1}{\sqrt{1-x^2}}$	$\displaystyle F(x)=-\cos^{-1}{(x)}+C$
$\displaystyle f(x)=\tan^{-1}\!{(x)}$	$\displaystyle f^{\,\prime}(x)=\frac{1}{1+x^2}$	$\displaystyle f(x)=\frac{1}{1+x^2}$	$\displaystyle F(x)=\tan^{-1}{(x)}+C$
$\displaystyle f(x)=\cot^{-1}\!{(x)}$	$\displaystyle f^{\,\prime}(x)=-\frac{1}{1+x^2}$	$\displaystyle f(x)=\frac{1}{1+x^2}$	$\displaystyle F(x)=-\cot^{-1}{(x)}+C$
$\displaystyle f(x)=\sec^{-1}\!{(x)}$	$\displaystyle f^{\,\prime}(x)=\frac{1}{x\,\sqrt{x^2-1}}$	$\displaystyle f(x)=\frac{1}{x\,\sqrt{x^2-1}}$	$\displaystyle F(x)=\sec^{-1}{(x)}+C$
$\displaystyle f(x)=\csc^{-1}\!{(x)}$	$\displaystyle f^{\,\prime}(x)=-\frac{1}{x\,\sqrt{x^2-1}}$	$\displaystyle f(x)=\frac{1}{x\,\sqrt{x^2-1}}$	$\displaystyle F(x)=-\csc^{-1}{(x)}+C$

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Created: Tuesday, 25 August 2020 02:28 EDT Last Modified: Last Modified: Monday, 10 January 2022 - 05:58 (EST)