Evaluate the indefinite integral.

$($ln$($x$))23$x

$$dx

Step $1$

We must decide what to choose for u$.$

If

u $=$ f$($x$),$

then

du $=$ f$\text{'}($x$)$ dx$,$

and so it is helpful to look for some expression in

$($ln$($x$))23$x

$$dx

$=$

$($ln$($x$))23$

$1$x

$$dx

for which the derivative is also present, though perhaps missing a constant factor.

For example,

ln$($x$)$

is part of this integral, and the derivative of

ln$($x$)$

is

$$$1$x$$

$,$ which is also present.

Step $2$

If we choose

u $=$ ln$($x$),$

then

du $=$

$1$x

$$dx$.$

If

u $=$ ln$($x$)$

is substituted into

$($ln$($x$))23$x

$$dx

$,$

then we have

$($ln$($x$))23$x

$$dx

$=$

u$23$

$1$x

$$dx

$.$

We must also convert

$1$x

$$dx

into an expression involving u$,$ but we know already that

$1$x

$$dx $=$

$($ln$($x$))88+$C

du$.$