Integration by parts
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intlib::byparts(integral
, du
)
intlib::byparts(integral, du)
performs on integral
the
integration by parts, where du
is the part to be
integrated and returns an expression containing the unevaluated partial
integral.
Mathematically, the rule of integration by parts is formally defined for indefinite integrals as
and for definite integrals as
.
intlib::byparts
works for indefinite as well
as for definite integrals.
If MuPAD^{®} cannot solve the integral for du
in
case of definite integration, the function call is returned unevaluated.
The first argument should contain a symbolic integral of type "int"
.
Such an expression can be obtained with hold
or freeze
(cf. Example 1).
The second argument du
should typically be
a partial expression of the integrand in integral
.
As a first example we apply the rule of integration by parts
to the integral
.
By using the function hold
we
ensure that the first argument is of type "int"
:
intlib::byparts(hold(int)(x*exp(x), x = a..b), exp(x))
In this case the ansatz is chosen as and thus v(x) = x.
In the following we give a more advanced example using the method
of integration by parts for solving the integral
.
For this we have to prevent that the integrator already evaluates
the integrals. Thus we first inactivate the requested integral with
the function freeze
F := freeze(int)(exp(a*x)*sin(b*x), x)
and apply afterwards partial integration with :
F1 := intlib::byparts(F, exp(a*x))
This result contains another symbolic integral, which MuPAD can solve directly:
eval(F1)
Here we demonstrate the difference between indefinite and definite integration by parts. If in the indefinite case the partial part cannot be solved, simply the unevaluated integral is plugged into the integration rule:
intlib::byparts(hold(int)(x*f(x), x),f(x))
This is no longer true for the definite case:
intlib::printWarnings(TRUE): intlib::byparts(hold(int)(x*f(x), x=a..b),f(x))
Warning: No closed form for 'int(f(x), x)' is found. [intlib::byparts]

Integral: an arithmetical
expression containing a symbolic 

The part to be integrated: an arithmetical expression 
Arithmetical expression.