Integration
The reverse process of differentiation, where a derived equation is restored to its original equation. There is a loss when differentiating an equation (i.e., constants are stripped away), which is why integration is only able to add an arbitrary constant $C$ or $K$ to the restored equation.
Has a notation of $\int f(x) \space dx$, read as “the integral of $f(x)$ with respect to $x$”.
- $f(x)$ is known as the integrand;
- $F(x)$ is known as anti-derivative of $f(x)$
- $C$ an arbitrary constant of integration.
# Rules
As with differentiation, integration has several rules that are the opposite of their counterparts with differentiation. These rules include the:
- power rule;
- reciprocal rule.
- sum and difference rule;
- constant multiple rule; and
- product rule.
# Power rule
$$ \int x^n \space dx = \frac {x^{n + 1}} {n + 1} + C $$
If the integral equation is in the form $(ax + b)^n$, the following formula can be used: $$ \int (ax + b)^n = \frac {(ax + b)^{n + 1}} {a(n + 1)} $$
# Reciprocal rule
When the exponent of $x$ is $-1$, it’s not possible to use the power rule as it’ll leave with a fraction dividing by zero. In this case, this rule is used instead:
$$ \int \frac 1 x \space dx = \int x^{-1} \space dx = ln \mid x \mid $$
# Sum and difference rule
$$ \int [f(x) + g(x)] \space dx = \int f(x) \space dx + \int g(x) \space dx $$ $$ \int [f(x) - g(x)] \space dx = \int f(x) \space dx - \int g(x) \space dx $$
# Constant multiple rule
$$ \int kf(x) \space dx = k \int f(x) \space dx $$