Integration

Last updated May 21, 2023

• Mathematics
• Calculus

The reverse process of differentiation

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.

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.

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

If the integral equation is in the form $(ax + b)^n$, the following formula can be used: $$\int (ax + b)^n = \frac {(ax + b)^{n + 1}} {a(n + 1)}$$

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:

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

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

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