Category: Antiderivative and Indefinite Integral

Concept of Antiderivative and Indefinite Integral

In the Calculus I (Differential Calculus) section, our main purpose was to find the derivative of a given function.

But often we need to solve the inverse task: given a function ${f{{\left({x}\right)}}}$, we need to find the function ${F}{\left({x}\right)}$ whose derivative is ${f{{\left({x}\right)}}}$. In other words, we need to find a function ${F}{\left({x}\right)}$ such that ${F}'{\left({x}\right)}={f{{\left({x}\right)}}}$.

Properties of Indefinite Integrals

Following properties of indefinite integrals arise from the constant multiple and sum rules for derivatives.

Property 1. If $a$ is some constant then $\int{a}\cdot{f{{\left({x}\right)}}}{d}{x}={a}\int{f{{\left({x}\right)}}}{d}{x}+{C}$. In other words cosntant can be factored out of integral sign.

Table of Antiderivatives

Below is a short list of functions and their general antiderivatives (we will give more complete table later).

Note that $\int{f{{\left({x}\right)}}}{d}{x}={F}{\left({x}\right)}+{A}$ and $\int{g{{\left({x}\right)}}}{d}{x}={G}{\left({x}\right)}+{B}$ where ${A}$ and ${B}$ are arbitrary constant.

Area Problem

Suppose that we are given continuous function $y={f{{\left({x}\right)}}}$ on ${\left[{a},{b}\right]}$ such that ${f{{\left({x}\right)}}}\ge{0}$ for all ${x}\in{\left[{a},{b}\right]}$.

We want to find area ${S}$ that lies under curve ${f{{\left({x}\right)}}}$ and bounded by lines ${x}={a}$, ${x}={b}$ and x-axis.