Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 11

The arc length of a function of x, f(x), over an interval is determined by the formula below:
L=int_a^bsqrt(1+((dy)/(dx))^2)dx
So using the function given, let us first find (dy)/(dx):
d/(dx)(ln(sin(x)))=(1/(sin(x)))*(cos(x))=(cos(x))/(sin(x))=cot(x)
We can now substitute this into our formula above:
L=int_a^bsqrt(1+((dy)/(dx))^2)dx=int_(pi/4)^((3pi)/4)sqrt(1+(cot(x))^2)dx
Which can then be simplified to:
L=int_(pi/4)^((3pi)/4)sqrt(1+cot^2(x))dx=int_(pi/4)^((3pi)/4)sqrt(csc^2(x))dx=int_(pi/4)^((3pi)/4)csc(x)dx
Then you find the definite integral as you normally would. (Using the method shown on the link below, you can find the integral of csc(x).)
L=int_(pi/4)^((3pi)/4)csc(x)dx=-ln|csc(x)+cot(x)|_(pi/4)^((3pi)/4)
L=-ln|csc((3pi)/4)+cot((3pi)/4)|-(-ln|csc(pi/4)+cot(pi/4)|)
L=-ln|sqrt(2)+(-1)|-(-ln(sqrt(2)+1|)=-ln|sqrt(2)-1|+ln|sqrt(2)+1|
Here, we will switch the two natural logarithm terms and use the quotient property to combine them into a single log:
L=ln|sqrt(2)+1|-ln|sqrt(2)-1|=ln|(sqrt(2)+1)/(sqrt(2)-1)|
If you rationalize the denominator (by multiplying by the conjugate and simplifying) and use the power property of logs, you are left with:
L=ln|(sqrt(2)+1)^2/1|=ln|(sqrt(2)+1)^2|=2ln|sqrt(2)+1|
So the exact value of the arc length of the graph of the function over the given interval is 2ln|sqrt(2)+1|
which is approximately equal to 1.76.

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