Integration

Integration is the reverse of differentiation. While differentiation finds the gradient of a function, integration finds the area under a function.

Rules of integration

Basics

Increase power, bring down power, differentiate inside

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

Why do we $+ c$ ?

Since any constant is removed in differentiation, for example:

\[ \frac{d}{dx}[3x + 5]dx = 3 \]

any constant would produce the same derivative.

\[ \frac{d}{dx}[3x + 5]dx = \frac{d}{dx}[3x + 6]dx = \frac{d}{dx}[3x + 999]dx = 3 \]

Therefore, we have to add the constant $c$ to represent an unknown constant.

Trigonometric functions

$y$	$\int{y}dx$
$sin(f(x))$	$\frac{cos(f(x))}{f'(x)} + c$
$cos(f(x))$	$$\frac{sin(f(x))}{f'(x)} + c$
$sec^2(f(x))$	$\frac{tan(f(x))}{f'(x)} + c$

For other integrals, make use of trigonomic identities to simplify the integrals before integration

\[ \int{sin^2(x)}dx = \int{\frac{1}{2}(1 - cos(2x))}dx = \frac{1}{2}\int{1 - cos(2x)}dx \]

Identity: $$ cos(2x) = 1 - sin^2(x) $$

\[ \int{cos^2(x)}dx = \int{\frac{1}{2}(cos(2x) + 1)}dx = \frac{1}{2}\int{cos(2x) + 1}dx \]

Identity: $$ cos(2x) = 2cos^2(x) - 1 $$

Exponential functions

\[ \int{e^{f(x)}}dx = \frac{e^{f(x)}}{f'(x)} + c \]

Definite integrals

Definite integrals are the integral of a function with a specified range. They are represented like this:

\[ \int_a^b{f(x)}dx = [F(x)]_a^b = F(b) - F(a) \]

\[ a = \textrm{lower bound} \]

\[ b = \textrm{upper bound} \]

$a$ is usually smaller than $b$

Finding area using definite integrals

1. Area between positive function and x axis

\[ \int_a^b{f(x)}dx \]

2. Area between negative function and x axis

\[ |\int_a^b{f(x)}dx| \]

We take the absolute value as the integral of a function under the x axis is negative, but area can only be positive.

3. Area with respect to y axis

Sometimes it is not possible to find the area using the x axis, or it is easier to find with respect to y axis.

In this case, we need to convert the function to represent x in terms of y.

For example:

\[ y = mx + c \]

will become

\[ x = \frac{y - c}{m} \]

Then, we can find the area similar to parts 1 and 2.

\[ \int_a^b{f(y)}dy \]

4. Area between 2 functions

What if we want to find the area between two curves $f(x)$$ and $g(x)$?

We can do this by taking the integral of the top function subtracted by the bottom function:

\[ \int_a^b{f(x) - g(x)}dx \]

This would always result in a positive value, even if both functions are in the negative x/y axis.

The same can be done with respect to y:

\[ \int_a^b{f(y) - g(y)}dy \]

Kinematics

We can use differentiation and integration to help us solve some kinematics related questions.

This is because some formulas in kinematics are related to each other:

Displacement (distance) is $s$
Velocity (distance over time)

\[ v = \frac{ds}{dt} \]

Acceleration (velocity over time)

\[ a = \frac{dv}{dt} \]

\(y\)	\(\int{y}dx\)
\(sin(f(x))\)	\(\frac{cos(f(x))}{f'(x)} + c\)
\(cos(f(x))\)	$\(\frac{sin(f(x))}{f'(x)} + c\)
\(sec^2(f(x))\)	\(\frac{tan(f(x))}{f'(x)} + c\)