00:01
We're given a curve and we're asked to find the exact length of this curve.
00:05
The curve has the equation y equals the natural log of the sicken of x, and we have that x lies between 0 and pi over 4.
00:25
Using this equation, we have that the derivative of y with respect to x is going to be 1 over seekin of x times the derivative of sicken of x, which we know to be sikin x tangent x.
00:38
So we get sicken of x times tangent of x over siquin of x.
00:50
And of course the sicken of x will cancel out since we have the x is non -zero, second of x is non -zero on the interval from zero to pi over four.
01:02
So this is simply going to be tangent of x.
01:10
And therefore, 1 plus d -y -d -x squared is the same as one plus tantal.
01:19
Tangent squared of x, which from trigid entities, this is the same as secant squared of x.
01:27
And so it follows that the length of the curve is the integral from x equals 0 to pi over 4 of the square root of the expression secan squared of x d x...