Indefinite Integration

An antiderivative of a function f (x) is any function g(x) whose derivative is f (x). If g(x) is an antiderivative of f (x), then g(x) + C is another antiderivative. In fact, every antiderivative is of the form g(x) + C for some constant C.

The indefinite integral of f (x) is the family of all antiderivatives of f (x) and is denoted $\int$f (x) dx.

$\blacktriangleright$ To evaluate an indefinite integral

1.
Place the insertion point anywhere in the expression.

2.
Choose Evaluate, press CTRL + E, or click itbpF0.3009in0.3009in0.0701inevaluate.wmf.

$\blacktriangleright$ Evaluate

$\dint$$\left(\vphantom{ 2x^{2}+3x+5}\right.$2x2 + 3x + 5$\left.\vphantom{ 2x^{2}+3x+5}\right)$ dx = ${\frac{{2}}{{3}}}$x3 + ${\frac{{3}}{{2}}}$x2 + 5x

You can evaluate indefinite integrals of piecewise-defined functions. The result will be a piecewise-defined function.


\begin{example}
Make the definition
\begin{displaymath}
f(x)=\left\{
\begi...
...f{Evaluate} will return the following piecewise-defined function.
\end{example}

$\displaystyle \int$f (x)dx = $\displaystyle \left\{\vphantom{
\begin{array}{ccc}
\frac{1}{2}x^{2} & \func{if} & x\leq 0 \\
x^{3} & \func{if} & 0<x
\end{array}
}\right.$$\displaystyle \begin{array}{ccc}
\frac{1}{2}x^{2} & \func{if} & x\leq 0 \\
x^{3} & \func{if} & 0<x
\end{array}$



Subsections