Anti-Differentiation

Integration

Anti-differentiation is a process that reverses differentiation, it is also known as integration. Essentially, when you integrate a the derivative of a function, you produce a family of curves in which the original function resides. This is so because constants differentiate to zero, thus we are unable to determine the original constant.

After we integrate a function, we are left with a function with some constant \(c\). This is named the 'constant of integration', and can be any number. We can substitute a previously known point of the original function to find this constant if needed.

\(\int_{}^{}f'(x)dx = f(x)+c\)

There are two forms of integration (anti-differentiation):

1. Indefinite Integral: \(\int_{}^{}f(x)dx\)


2. Definite Integral : \(\int_{a}^{b}f(x)dx\)

Both forms feature the exact same procedure, except that definite integrals have one additional step, we'll focus on indefinite integrals and the integration process for now.

Indefinite integrals

Indefinite integrals are notated with: \(\int_{}^{}f(x)dx\), in which \(f(x)\) can be any funtion of any variable \(v\): as long as the integral ends with \(dv\).

That long spiny thing at the start of the integral indicated a 'sum of', followed by some function of \(v\), followed by \(dv\). The long thing and the \(dv\) act as bookmarks indicating the start and end of the function. Of course, \(dv\) also means 'a change of v'. So essentially the integral defines the sum of the changes of v.

Rules for integration

These rules are simply the reverse of differentiation rules. This will make more sense with practice:

#1: Integrating constants

Simply multiply the constant by the integrating variable (essentially the integration chain rule), and as with all indefinate integrals, add a constant \(c\): we don't know what \(c\) is, so just leave it so.

#2: The power rule for integration

This is another application of the chain rule for integration.

\(\int ax^ndx = \cfrac{ax^{n+1}}{n+1} + c\)

1) \(\int 4t^3 + 4x + 8dt\)

\(\begin{aligned} &= \cfrac{4t^4}{3+1} + \cfrac{4t^2}{1+1} + 8t + c \\ \\ &= t^4 + 2t^2 + 8t + c \end{aligned} \)

#3: Linearity of integration

Similar to differentation, an integral involving a plus or minus is equal to the integral of each term with the previous plus/minus sign:

\(\int( a(x) - b(x) + c(x)dx = \int a(x))dx - \int b(x)dx + \int c(x)dx \)

Now, before you can apply this rule you must know that it is possible to take out a multiple from an integral (aswell with a derivative):

\(\int ax^2 + ax dx = a\int x^2 + x dx\)

You cannot however, take out any variables.

1) Simplify \(\cfrac{1}{3} \int 3x^2 + 9x dx \)

\(\begin{aligned} &= \int \cfrac{1}{3} (3x^2 + 9x)dx \\[5pt] &= \int x^2 + 3x dx \\[5pt] &= \cfrac{x^3}{3} + \cfrac{3x^2}{2} + c \end{aligned} \)

#3: Integrating the chain rule

Again, just the reverse of the chain rule. Again, confusing at first, but the more practice you do the easier it will be to wrap your head around it.

\(\int f'(x)[f(x)]dx = \cfrac{[f(x)]^{n+1}}{n+1} + c\)

Please note that this rule only applies if \(f(x)\) is linear. If not, you must expand \(f(x)\) and then apply the power rule to each term.

Making use of linearity of integration and the ability to take out/insert factors within the integral, we can manipulate the integral such that it fits the form above. From thereon, follow through the formula to integrate using the chain rule.

Finding families of curves with the same derivative function

The main application of indefinite integration is finding a function based on limited information, such as it's derivative and a co-ordinate from the original function. This can be achieved as previously described, using an indefinite integral to calculate the function, then subbing a known value to find \(c\).

Just like that. By the way, if you didn't know, those three dots in a triangle at the last line mean 'therefore', good to use as a way to declare a final statement.