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(x)` find function `F(x)` whose derivative is `f(x)`. In other words, we need to find function `F(x)` such that `F'(x)=f(x)` .

Properties of Indefinite Integrals

Following properties of indefinite integrals arise from the Constant Multiply and Sum rules for derivatives.

Property 1. If `a` is some constant then `int a*f(x)dx=a int f(x)dx +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(x)dx=F(x)+A` and `int g(x)dx=G(x)+B` where `A` and `B` are arbitrary constant.

Area Problem

Suppose that we are given continuous function `y=f(x)` on `[a,b]` such that `f(x)>=0` for all `x in [a,b]`.

We want to find area `S` that lies under curve `f(x)` and bounded by lines `x=a`, `x=b` and x-axis.