 2  Limits and Their Properties
 3  Differentiation
 4  Applications of Differentiation
 5  Integration
 7  Applications of Integration
 MAT 280: Homework
 MAT 280: Exams
 MAT 280: Sandbox
 MAT 280: Challenge
 MAT 280: Quizzes
 Differentitation Rules Reference Page
 Sum/Difference Rule:
$$
\int [f(x) \pm g(x)] \; dx= \int f(x) \; dx \pm \int g(x) \; dx
$$  Constant Multiplier Rule:
$$
\int k \cdot f(x) \; dx= k \int f(x)\; dx
$$
Let \(C\) be a constant.

\( \displaystyle
\int 0 \; dx = C
\) 
\( \displaystyle
\int \; dx = x + C
\) 
\( \displaystyle
\int x^n \; dx= \begin{cases}
\dfrac{x^{n+1}}{n+1} +C , & \; n \not = 1 \\
\\
\ln x + C, & \; n = 1
\end{cases}
\) 
\( \displaystyle
\int \cos x \; dx = \sin x + C
\) 
\( \displaystyle
\int \sin x \; dx = \cos x + C
\) 
\( \displaystyle
\int \tan x \; dx = – \ln \cos x + C
\) 
\( \displaystyle
\int \cot x \; dx = \ln \sin x + C
\) 
\( \displaystyle
\int \sec^2 x \; dx = \tan x + C
\) 
\( \displaystyle
\int \csc^2 x \; dx = \cot x + C
\) 
\( \displaystyle
\int \sec x \tan x \; dx = \sec x + C
\) 
\( \displaystyle
\int \csc x \cot x \; dx = \csc x + C
\) 
\( \displaystyle
\int \csc x \; dx = \ln\csc x+\cot x + C
\) 
\( \displaystyle
\int \sec x \; dx = \ln\sec x + \tan x + C
\) 
\( \displaystyle
\int e^x \; dx = e^x+ C
\) 
\( \displaystyle
\int a^x \; dx = \dfrac{a^x}{\ln a} + C, \; \; a>0, a \not = 1
\)
Let \(g\) and \(u\) be functions such that \(f(x) = g(u(x))\) (That is, \(f(x)\) can be written as a composition of \(g\) with \(u\)) and assume that the antiderivative \(G(x)\) of \(g\) is known then
$$
\begin{aligned}
\int f(x) \; dx & = \int g(u(x)) \dfrac{du}{dx} \; dx \\
& = \int g(u) \; du = G(u) + C.
\end{aligned}
$$
$$
\begin{aligned}
\int f(x) \; dx & = \int g(u(x)) \dfrac{du}{dx} \; dx \\
& = \int g(u) \; du = G(u) + C.
\end{aligned}
$$
The Fundamental Theorem of Calculus (Part 1): If \(f(x)\) is continuous on \([a,b]\) and \(F(x)\) is an antiderivative of \(f(x)\) then
$$
\int_a^b f(x) \; dx = F(x)\Big_a^b
$$
$$
\int_a^b f(x) \; dx = F(x)\Big_a^b
$$
The Fundamental Theorem of Calculus (Part 2): If \(f\) is continuous on \([a,b]\) then
$$
\frac{d}{dx} \left[\int_a^x f(t) \; dt \right] = f(x).
$$
$$
\frac{d}{dx} \left[\int_a^x f(t) \; dt \right] = f(x).
$$
The average value of \(f(x)\) on \([a,b]\) is denoted \(f_{avg}\) and is given by
$$
f_{avg} = \frac{1}{ba}\int_a^b f(x)\; dx.
$$
$$
f_{avg} = \frac{1}{ba}\int_a^b f(x)\; dx.
$$
The definite integral of \(f(x)\) on \([a,b]\) is denoted
$$
\int_a^b f(x) \; dx
$$
and informally represents the signed “area” of the region “bounded” by the graphs of \(y = f(x)\), there vertical lines \(x=a\) and \(x =b\), and the \(x\)axis.
$$
\int_a^b f(x) \; dx
$$
and informally represents the signed “area” of the region “bounded” by the graphs of \(y = f(x)\), there vertical lines \(x=a\) and \(x =b\), and the \(x\)axis.