### Video Transcript

Write, but do not evaluate, an integral for the arc length of the curve π¦ is equal to the natural logarithm of two π₯ to the fifth power plus seven between π₯ is equal to one and π₯ is equal to four.

The question gives us a curve and it wants us to write an integral which gives us the arc length of this curve between the values of π₯ is equal to one and π₯ is equal to four. And we know how to find the arc length of certain curves. We know if π prime is a continuous function on the closed interval from π to π. Then the length of the curve π¦ is equal to π of π₯ between the values of π₯ is equal to π and π₯ is equal to π is equal to the integral from π to π of the square root of one plus π prime of π₯ squared with respect to π₯.

We want the arc length of the curve π¦ is equal to the natural logarithm of two π₯ to the fifth power plus seven between the values of π₯ is equal to one and π₯ is equal to four. So weβll set our function π of π₯ to be the natural logarithm of two π₯ to the fifth power plus seven. Weβll set π equal to one and π equal to four. Then, to use this arc length formula, we just need to show that π prime is continuous on the closed interval from π to π. So letβs find an expression for π prime of π₯.

We can see if we try to differentiate π of π₯, we need to differentiate a composite function. So weβll do this by using the chain rule. If we set π’ of π₯ to be our inner function two π₯ to the fifth power, then we can see that π is a function of π’ and π’ is a function of π₯. Then by the chain rule, the derivative of π of π’ of π₯ is equal to π prime evaluated at π’ of π₯ times π’ prime of π₯. Therefore, using the chain rule, we have π prime of π₯ is equal to the derivative of the natural logarithm of π’ with respect to π’ times the derivative of two π₯ to the fifth power with respect to π₯.

And we can evaluate each of these derivatives separately. The derivative of the natural logarithm of π’ with respect to π’ is one over π’. And the derivative of two π₯ to the fifth power with respect to π₯ is 10π₯ to the fourth power. Remember, this is an expression for π prime of π₯. So we should give our answer in terms of π₯. Weβll do this by using our substitution π’ is equal to two π₯ to the fifth power. Using this, we get 10π₯ to the fourth power divided by two π₯ to the fifth power, which simplifies to give us five over π₯. So we found that π prime of π₯ is equal to five over π₯. This is a rational function. And we know that all rational functions are continuous on their domain. In fact, we can see the domain of π prime of π₯ is all values of π₯ not equal to zero.

Remember, we wanted π prime to be continuous on our closed interval from one to four. This doesnβt contain π₯ is equal to zero. So our function π prime is continuous on this interval. Therefore, weβve justified the use of our integral formula to find the length of our arc. Substituting in π prime of π₯ as five over π₯, π is equal to one, and π is equal to four, we get the length of our arc is equal to the integral from one to four of the square root of one plus five over π₯ squared with respect to π₯. And the last thing weβll do is simplify five over π₯ all squared to give us 25 over π₯ squared.

Therefore, weβve shown we can find the arc length of the curve π¦ is equal to the natural logarithm of two π₯ to the fifth power plus seven between the values of π₯ is equal to one and π₯ is equal to four. By evaluating the integral from one to four of the square root of one plus 25 over π₯ squared with respect to π₯.