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