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 $\int f(x)dx$, 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

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$$

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:

$$\int \frac{1}{x}dx=\int x^{-1}dx=ln\mid x\mid$$

Sum and difference rule

$$\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$$ $$\int [f(x)-g(x)]dx=\int f(x)dx-\int g(x)dx$$

Constant multiple rule

$$\int kf(x)dx=k\int f(x)dx$$