Note: Use only in the MuPAD Notebook Interface. This functionality does not run in MATLAB. |
Integration by parts is one of the common methods for computing integrals. Using this method, you rewrite the original integral in terms of an expression containing a simpler integral. Integration by parts for indefinite integrals uses the definition:
.
For definite integrals, integration by parts is defined as follows:
.
Internally, MuPAD^{®} uses integration by parts along with
other integration methods. To use this method explicitly, call the intlib::byparts
function.
If you want to integrate an expression by parts, keep the original
integral unevaluated. By default, int
returns evaluated integrals. Use
the hold
or freeze
commands to prevent
evaluation of the integral:
f := freeze(int)(exp(a*x)*cos(I*x*b), x)
Call intlib::byparts
and
specify the part of an expression you want to integrate. For example,
specify
:
f_int := intlib::byparts(f, exp(a*x))
To evaluate the resulting integral, use the eval
command:
eval(f_int)
If the resulting expression is too long, try using the simplify
or Simplify
function:
Simplify(%)
Change of variable is also one of the common methods for computing
integrals. For explicit use of this method, MuPAD provides the intlib::changevar
function.
When changing an integration variable, you need to keep the integral
unevaluated. By default, int
returns
evaluated integrals. Use the hold
or freeze
commands to prevent
evaluation of the integral:
f := intlib::changevar(hold(int)(sin(exp(x)), x), t = exp(x), t)
To evaluate the resulting integral, use the eval
command:
eval(f)
The change of variable method also works for computing definite integrals:
f := intlib::changevar(hold(int)(x/sqrt(1 - x^2), x = a..b), t = x^2, t)