Notation Notation

Introduction

An antiderivative, when differentiated, gives the original function.
An integral is the value of the area under a curve for a domain, calculated using the antiderivative.

Antiderivative

\int f(x) \, dx = F(x) + C

Integral

\int_a^b f(x) \, dx = F(x)\Big|_a^b = F(b) - F(a)

Common Antiderivatives

\int x^n \, dx = \frac{1}{n+1}x^{n+1} + C

\int e^x \, dx = e^x + C

\int \sin(x) \, dx = -\cos(x) + C

\int \cos(x) \, dx = \sin(x) + C

\int \frac{1}{x}dx = \ln|x| + C

Examples

Example 1:
Find the area under

f(x) = 3x^2
on
[0,1]
.
\begin{align*} \int_0^1 3x^2 \, dx &= x^3 + C \Big|_0^1 \\\\ &= 1^3 - 0^3 \\\\ &= 1 \end{align*}


Example 2:
Find the area under
\sin(x)
on
[0,\pi]
.
\begin{align*} \int_0^\pi \sin(x) \, dx &= -\cos(x) + C \Big|_0^\pi \\\\ &= -\cos(\pi) - (-\cos(0)) \\\\ &= 1 + 1 \\\\ &= 2 \end{align*}


Example 3:
find the area under
\cos(x)
on
[0,\pi]
.
\begin{align*} \int_0^\pi \cos(x) \, dx &= \sin(x) + C \Big|_0^\pi \\\\ &= \sin(\pi) - \sin(0) \\\\ &= 0 - 0 \\\\ &= 0 \end{align*}