Since it looks like you have a parametric curve of r(t) =〈2t, ln(t), t2〉, we first need to find r'(t), then add the squares of all those derivatives, take the square root, and integrate from 1 to 3. We're basically looking for the change that the function undergoes at each instance and using the distance formula to find how much the length is in each instance. Then the integral is just adding up all those minutely small instances.
r'(t) =〈2, 1/t, 2t〉
Each individual distance is √(4 + 1/t2 + 4t2)dt. To simplify this square root, we can multiply everything by t2/t2 and we get √[(4t4 + 4t2 + 1)/t2] = (2t2 +1)/t. Now we simply integrate.
∫13 2t + 1/t dt
t2 + ln(t) |13
8 + ln(3)
So the length of the curve is 8 + ln(3).
