Step By Step Calculus » 15.1 - Indefinite Integrals

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Synopsis
Now we want to reverse the differentiation process: Given a derivative ff we want to find another function FF such that F'(x)=f(x)F'(x)=f(x). This FF is called an antiderivative for f.f. Two antiderivatives for ff can differ by at most a constant. Therefore, if FF is one antiderivative for f,f, then the general antiderivative is denoted F(x)+cF(x)+c and the set of antiderivatives is \{F(x)+c: c\in\mathbb{R}\}.\{F(x)+c: c\in\mathbb{R}\}.
The Indefinite integral of a function f(x)f(x) is the antiderivative of f(x)f(x) in the largest possible domain. It is denoted by
\displaystyle \int f(x) dx
\displaystyle \int f(x) dx
and is read as “the indefinite integral of f(x)f(x) with respect to x". The function ff that is being integrated is called the integrand, and the variable xx is called the variable of integration.
The Derivative Linearity rule \displaystyle \frac{d}{dx}\left(aF(x)+bG(x)\right)=a\frac{d}{dx}F(x)+b\frac{d}{dx}G(x)=af(x)+bg(x)\displaystyle \frac{d}{dx}\left(aF(x)+bG(x)\right)=a\frac{d}{dx}F(x)+b\frac{d}{dx}G(x)=af(x)+bg(x) implies the linearity rule for antiderivative, i.e.

\displaystyle { \textbf{(Linearity Property)}\qquad \int [af(x)+bg(x)]dx=a\int f(x)dx+b\int g(x)dx. }\displaystyle { \textbf{(Linearity Property)}\qquad \int [af(x)+bg(x)]dx=a\int f(x)dx+b\int g(x)dx. }


Finding an antiderivative for a given function is a bit of an art from based upon the differentiation rules. We will concentrate on the simple cases herein and build the rules for complex cases in the next few sections.
When dealing with piecewise defined functions, finding the indefinite integral or antiderivative the key idea is to remember is that an antiderivative has to be differentiable which implies continuity. We can use the following steps to determine the indefinite integral or antiderivative of a piecewise defined function.
Step 1: (Check integrand continuity)Ensure that f(x)f(x) is continuous over its whole domain.
Step 2: (Find piecewise antiderivatives)Find antiderivatives of each piece separately.
Step 3: (Use Single Constant)Establish relation between the constants in each piece using the concept of continuity.