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To integrate a mathematical expression
to find an expression
F such that the first derivative
f. The expression
an antiderivative of
f. Integration is a more complicated
task than differentiation. In contrast to differentiation, there is
no general algorithm for computing integrals of an arbitrary expression.
When you differentiate an expression, the result is often represented
in terms of the same or less complicated functions. When you integrate
an expression, the result often involves much more complicated functions
than those you use in the original expression. For example, if the
original expression consists of elementary functions, you can get
the result in terms of elementary functions:
int(x + 1/(x^2), x)
The following integrand also consists of standard trigonometric functions, but here the integrator cannot return the result in terms of elementary functions. The antiderivative involves a special function:
When you compute an indefinite integral, MuPAD® implicitly
assumes that the integration variable is real. The result of integration
is valid for all real numbers, but can be invalid for complex numbers.
You also can define properties of the integration variables by using
The properties you specify can interfere with the assumption that
the integration variable is real. If MuPAD cannot integrate an
expression using your assumption, the
int function issues a warning. Use the
to switch the warnings on and off. For example, switch on the warnings:
Suppose you want to integrate the following expression under
the assumption that the integration variable is positive. This assumption
does not conflict with the assumption that the variable is real. The
int command uses your
f := abs(x): int(f, x) assuming x > 0
Integrate this expression under the assumption that
an integer. MuPAD cannot integrate the expression over a discrete
subset of the real numbers. The
int command issues a warning, and then
integrates over the field of real numbers:
int(f, x) assuming x in Z_
Warning: Unable to integrate when 'x' has property 'Z_'. Using assumption 'x' has property 'R_' for integration. [intlib::int]
For a discrete set of values of the integration variable, compute a sum instead of computing an integral. See Summation for details.
Now integrate under the assumption that
cannot compute the integral of the expression over imaginary numbers.
It issues a warning and integrates the expression over the domain
of complex numbers:
assume(x, Type::Imaginary); int(f, x)
Warning: Unable to integrate when 'x' has property 'Dom::ImageSet(x*I, x, R_)'. Using assumption 'x' has property 'C_' for integration. [intlib::int]
For more information about the assumptions, see Properties and Assumptions. Before you proceed with
other computations, clear the assumption on the variable
Also, disable the warnings: