Question

If $\int {\left( {u\dfrac{{dv}}{{dx}}} \right)} \;dx = uv - \int {wdx} ,$ then W=
1)$\dfrac{{du}}{{dx}} \cdot \dfrac{{dv}}{{dx}}$
2)$v.\dfrac{{du}}{{dx}}$
3)$\dfrac{d}{{dx}}\left( {uv} \right)$
4)$\dfrac{d}{{dx}}\left( {\dfrac{u}{v}} \right)$

Answer 1

Hint: Use the integral function assuming that u as one function and $\dfrac{{dv}}{{dx}}$ as another function try to match and select for integral of $\int {ab\;dx} $

Complete step-by-step answer:
 Let’s begin with what type of Integral function is given. The function given is $\int {\left( {u\dfrac{{dv}}{{dx}}} \right)dx} $
As we can see that form is 2 functions or variables which is u and second is dv / dx.
As we match we integral property we can equals this in the form of
\[\int {a.b\;dx\; = a} \;\int {dx\; - \;\int {\left( {\dfrac{{da}}{{dx}}\smallint ab\;dx} \right)} } dx\]
Let’s assume $a$ as u and b as $\dfrac{{dv}}{{dx}}$
By putting the values we get.
$\int {\left( {u\dfrac{{dv}}{{dx}}} \right)} \;dx = u\int {\dfrac{{dv}}{{dx}}.dx - \int {\left( {\dfrac{{du}}{{dx}}\smallint \dfrac{{dv}}{{dx}}dx} \right)} } \;dx$
$\int {\left( {u\dfrac{{dv}}{{du}}} \right)} dx = uv - \int {\left( {\dfrac{{dv}}{{dx}} \cdot v} \right)} dx$
So if we compare the equation with the given one we get.
$w = \dfrac{{du}}{{dx}} \cdot v$

So, option 2 is the correct option

Note: The basic properties of Integration and differentiation is the basic source to solve this type of question matching the correct part from the option.