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Section 4.1 Intro to Integrals

Definition 4.1.1.

A function \(F(x)\) is called an antidervative of \(f(x)\) on an open interval if:

\begin{equation*} \dfrac{dF}{dx} = \dfrac{d}{dx}F(x) = F'(x) = f(x) \end{equation*}

for all \(x\) in the interval.

\(F(x) = 3x^2\) is an antiderivative of \(f(x) = 6x\text{.}\)

The process of finding the antiderivative is called antidifferentiation or integration.

Consider, \(\dfrac{d}{dx}(F(x)) = f(x)\) to integrate \(f(x)\) we do the following:

\begin{equation*} \int f(x) dx = F(x) + c \end{equation*}

This means the same as:

\begin{equation*} \dfrac{d}{dx}(F(x)) = f(x) \end{equation*}

but we know \(f(x)\) and we are finding \(F(x)\text{.}\)

The equation

\begin{equation*} \int f(x) dx = F(x) + c \end{equation*}

would be said as:

"The integral of \(f(x)\) with respet to \(x\) is \(F(x)\) plus \(c\text{.}\)"

Note 4.1.3.

Assume \(c\) is a constant, \(\dfrac{d}{dx}(c) = 0\) so,

\begin{align*} \dfrac{d}{dx}(F(x) + c) \amp = \dfrac{d}{dx}(F(x)) + \dfrac{d}{dx}(c)\\ \amp = f(x) + 0 \\ \amp = f(x) \end{align*}

This means \(F(x) + c\) represesnts all possible function that share \(f(x)\) as their derivative.

If \(f(x) = 5\text{,}\) what is \(F(x)\text{?}\)

\begin{equation*} \dfrac{d}{dx}\text{( ? )} = 5 \end{equation*}
\begin{equation*} 5x + c = ? \end{equation*}

\(c\) lets us shift the value of the function vertically without changing the function.

We call this an indefinite integral (finding the set of all functions that share a common derivative).

Example 4.1.1.

Find \(\displaystyle\int 3 dx\)

\begin{align*} \int 3 dx \amp = F(x) + c\\ \amp = 3x + c \end{align*}

Since \(\dfrac{d}{dx} \left( 3x + c \right) = 3\)

Note 4.1.4.

The \(c\) allows us to write all related functions generally.

Here are a list of common integrals:

  • \(\displaystyle \displaystyle\int dx = \displaystyle\int 1 dx = x + c \)

  • \(\displaystyle \displaystyle\int x dx = \dfrac{1}{2} x^2 + c \)

  • \(\displaystyle \displaystyle\int x^2 dx = \dfrac{1}{3} x^3 + c \)

  • \(\displaystyle \displaystyle\int \cos(x) dx = \sin(x) + c \)

  • \(\displaystyle \displaystyle\int \sin(x) dx = - \cos(x) + c \)

  • \(\displaystyle \displaystyle\int \sec^2(x) dx = \tan(x) + c \)

  • \(\displaystyle \displaystyle\int \csc^2(x) dx = - \cot(x) + c \)

  • \(\displaystyle \displaystyle\int \sec(x)\tan(x) dx = \sec(x) + c \)

  • \(\displaystyle \displaystyle\int \csc(x)\cot(x) dx = - \csc(x) + c \)

Power Rule for Integrals.

The power rule for integrals states

\begin{equation*} \int x^r dx = \dfrac{1}{r + 1} \cdot x^{r + 1} + c \end{equation*}

for all \(r \neq -1\text{.}\)

Let \(k\) and \(c\) be constants, \(F(x)\) and \(G(x)\) be antiderivatives of \(f(x)\) and \(g(x)\text{.}\) Then all of the following are true:

  • \begin{equation*} \int k \cdot f(x) dx = k \int f(x) dx = k \cdot F(x) + c \end{equation*}

    Note 4.1.5.

    Since \(c\) is unknown \(k \cdot c\) doesn't have any meaning, so we can still call it \(c\text{.}\)

  • \begin{align*} \int f(x) \pm g(x) dx \amp = \left( \int f(x) dx \right) \pm \left( \int g(x) dx \right)\\ \amp = \left( F(x) + c \right) \pm \left( G(x) + c \right) \\ \amp = F(x) \pm G(x) + c \end{align*}

Example 4.1.2.

Let's put some of these rules together:

\begin{align*} \int 3x^2 - \cos(x) dx \amp = \int 3x^2 dx - \int \cos(x) dx\\ \amp = 3 \int x^2 dx - \int \cos(x) dx \\ \amp = 3\left( \dfrac{1}{3} x^3 \right) - \left( \sin(x) \right) + c\\ \amp = x^3 - \sin(x) + c \end{align*}