Category: Antiderivative and Indefinite Integral

Concept of Antiderivative and Indefinite Integral

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

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