Evaluate

$16$ x$2$

x$2$

$$dx$.$

Solution

Let

x $=4$ sin$(),$

where

$2$

$$

$2$

$.$

Then

dx $=$

d$$

and

$16$ x$2$

$=$

$1616$ sin$2()$

$=$

$16$ cos$2()$

$=4|$cos$()|=4$ cos$().$

$($Note that

cos$()0$

because

$2$

$$

$2$

$.)$

Thus, the inverse substitution rule gives

$16$ x$2$

x$2$

$$dx

$=$

$4$ cos$()16$ sin$2()$

$$

$$d$$

$=$

cos$2()$

$$d$$

$=$

cot$2()$ d$$

$=$

$($csc$2()1)$ d$$

$=$

$+$ C$.$