Integration by Parts

If we have studied the integral substitution method consisting of two versions, it is incomplete if we do not study the partial integral method. Because both of these methods are often used in solving integral problems both integral and uncertain integrals.

If integration uses substitution fails, we might be able to use double substitution, which is better known as partial integral. This method is based on the integration of the formula for derivatives of the product of two functions.

Suppose that $latex u = u (x) $ and $latex v = v (x) $, then:
$latex D_x [uv] = uv '+ vu' $
or
$latex uv '= D_x [uv] -vu' $

By integrating the two segments of the equation, we obtain
$latex \int uv ' \ dx = uv - \int vu' \ dx $

Because $latex dv = v ' \ dx $ and  $latex du = u' \ dx $, the above equation is usually written with the symbol as follows.

$$ \int u \ dv = uv - \int v \ du $$

The formulas that correspond to the integral of course are:

$$ \int_{a}^{b} u \ dv = [uv]_{a}^{b} - \int_{a}^{b} v \ du $$

Question example:
Look for $latex \int x \cos x \ dx $

Settlement:

The question cannot indeed be done by using a substitution method. We want to write $latex x \cos x \ dx $ as $latex u \ dv $. The possibility is to suppose that $latex u = x $ and $latex dv = \cos x \ dx $ or $latex u = \cos x $ and $latex dv = x \ dx $. We must be smart to choose between the two which are more appropriate. To determine which one has to be $latex u $ and $latex dv $, you can read the way in another article titled Select u and dv in Integral Partial, I can.

For example, $latex u = x $ and $latex dv = \cos x \ dx $
Then,
$latex du = dx $ and
$$ \begin{align} v & = \int \cos x \ dx \ & = \sin x \end{align} $$
So that,
$$ \begin{align} & \int \underbrace{x}_u \underbrace{cos x dx}_{dv} \\ & = \underbrace{x}_{u} \underbrace{sin x}_{v} - \int \underbrace{sin x}_{v} \underbrace{dx}_{du} \\ & = x \sin x - (- \cos x) + C \\ & = x \sin x + \cos x + C \end{align} $$