CPM Homework Help : APCALC Problem 12-108

Multiple Choice: Which of the following integrals represents the arc length of the curve $y = \ln(\cos(x))$ from $x = 0$ to $x =\frac { \pi } { 3 }$ ?

$\int _ { 0 } ^ { \pi / 3 } \sqrt { 1 + \operatorname { sec } ^ { 2 } ( x ) } d x$

$\int _ { 0 } ^ { \pi / 3 } \sqrt { x ^ { 2 } + ( \operatorname { ln } ( \operatorname { cos } ( x ) ) ) ^ { 2 } } d x$

$\int _ { 0 } ^ { \pi / 3 } \operatorname { sec } ( x ) d x$

$\int _ { 0 } ^ { \pi / 3 } \sqrt { 1 + \operatorname { tan } ( x ) } d x$

$\int _ { 0 } ^ { \operatorname { ln } ( 0.5 ) } \sqrt { 1 + \operatorname { tan } ^ { 2 } ( x ) } d x$

Step:

$y^\prime=-\frac{\sin(x)}{\cos(x)}=-\tan(x)$

Hint:

Now use the arc length formula.
Note: You will need to use one of the Pythagorean Identities to simplify your result.