| The rule for differentiating
a product of two differentiable functions leads to the integration
by parts formula.
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|
Let
f (x)
and g
(x)
are differentiable functions, then the product rule gives
|
| [ f
(x) g (x)]'
= f (x)
g (x)' + g (x)
f ' (x), |
| by
integrating both sides |
 |
|
| Since the integral of the derivative
of a function is the function itself, then
|
 |
|
and by rearranging obtained is
|
 |
|
| the integration by parts
formula.
|
| By substituting
u = f
(x) and
v
= g (x)
then, du =
f ' (x)
dx and dv
= g' (x)
dx, so that
|
 |
| To apply the above formula, the
integrand of a given integral should represent the product of one
function and the differential of the other. The selection of the function
u and the differential dv
should simplify the
evaluation of the remaining integral.
|
| In some cases it will be necessary to
apply the integration by parts repeatedly to obtain a simpler
integral.
|
