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Section 15.2 Indefinite Integrals and Antiderivatives
The
Int() and
int() commands may also be used for indefinite integrals. As mentioned in
Definite Integrals , the inert command
Int() displays the integral, and
int() evaluates the integral directly. Either command may be used for finding an antiderivative of a given function.
\begin{equation*}
\displaystyle \int \!\sin \left( x \right) {dx}
\end{equation*}
\begin{equation*}
\displaystyle -\cos \left( x \right)
\end{equation*}
Notice that Maple does not include the addition of the constant of integration
\(+C\) when evaluating indefinite integrals.
The inert
Int() command can be combined with the
value() command to display the integral symbolically and then compute the indefinite integral.
> p(x) := 1/sqrt(1 + x^2);
\begin{equation*}
\displaystyle p\, := \,x\mapsto {\frac {1}{\sqrt {{x}^{2}+1}}}
\end{equation*}
> Int(p(x), x); value(%);
\begin{equation*}
\displaystyle \int \! \left( \sqrt{{x}^{2}+1} \right) ^{-1}{dx}
\end{equation*}
Aside The function arcsinh
\(()\) is called the inverse hyperbolic sine function. Functions like
\(\cosh()\) and
\(\sinh()\) are called hyperbolic functions. These functions are analogs of the ordinary trigonometric, or circular, functions; just as the points
\((\cos(t), \sin(t))\) form a circle with a unit radius, the points
\((\cosh(t), \sinh(t))\) form the right half of the equilateral hyperbola.
\begin{equation*}
\displaystyle \arcsinh \left( x \right)
\end{equation*}
