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Integration by parts and differentials
A differential of a function y = f(x), is defined as dy = f’(x) dx, where f’(x) is the
derivative of f(x). Differentials are used to represent small increments in the
variables. The differential of a product of two functions, y = u(x)v(x), is given by
dy = u(x)dv(x) +du(x)v(x), or, simply, d(uv) = udv - vdu. Thus, the integral of udv
= d(uv) - vdu, is written as . Since by the definition of
a differential, ∫dy = y, we write the previous expression as
.
This formulation, known as integration by parts, can be used to find an integral
if dv is easily integrable. For example, the integral ∫xe
x
dx can be solved by
integration by parts if we use u = x, dv = e
x
dx, since, v = e
x
. With du = dx, the
integral becomes ∫xe
x
dx = ∫udv = uv - ∫vdu = xe
x
- ∫e
x
dx = xe
x
- e
x
.
The calculator provides function IBP, under the CALC/DERIV&INTG menu, that
takes as arguments the original function to integrate, namely, u(X)*v’(X), and
the function v(X), and returns u(X)*v(X) and -v(X)*u’(X). In other words, function
IBP returns the two terms of the right-hand side in the integration by parts
equation. For the example used above, we can write in ALG mode:
Thus, we can use function IBP to provide the components of an integration by
parts. The next step will have to be carried out separately.
It is important to mention that the integral can be calculated directly by using,
for example,
∫∫∫
−= vduuvdudv )(
∫∫
−= vduuvudv