Evaluate$$

$0$

sin$2(6$x$)$dx$.$

SOLUTION$$If we write$$

sin$2(6$x$)=1$cos$2(6$x$),$

$$the integral is no simpler to evaluate. Using the half$-$angle formula for$$

sin$2($x$),$

$$however$,$ we have

$0$

sin$2(6$x$)$dx$=$

$12$

$0$

$(1$cos$(12$x$))$dx

$=$

$12$

x$$

$$

$$

$0$

$$

$=$

$12$

$$

$$

$$

$$

$12$

$0$

$$

$$

$$

$=$

$$