In Exercises 39–48, use an appropriate substitution and then a...

Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ dy / (y√(1 + (ln y)²)) from 1 to e

Accepted Answer

Welcome back everyone. Find the value of the integral from 1 up to e to the power of 4 of d y divided by y square root of 16 + ln square y. A elan of 2 + square root of 3. Belon of 2 + square root of 2. Celan of 1 + square root of 3, and d l n of 1 + square root of 2. What we're going to do in this problem is use substitution. Let's suppose that U is equal to L of Y. The next step is to find the derivative of U with respect to Y, which is the derivative of L and of Y, and that's 1 divided by Y. Multiplying both sides by Dy, we can show that duu is equal to Dy divided by y, which is exactly what we have within the integral. So within the integrant we have the y divided by y, square root of 16 plus LN square root of y. Our next step is to replace the limits of integration, so the lower 1 u1 is going to be ln off 1, which is 0. And u2 is going to be equal to ln of e to the power of 4, which is 4. We're now ready to rewrite the integral in terms of U and DU, so that'll be the integral from 0 up to 4. DY divided by Y becomes DU. And in the denominator we have square root of 16 + u squared. What we can do is rewrite 16s 4 squared, right? It is a perfect square, and now we're going to remember one of the integrals, and that would be the integral of the u. Divided by square root of A2 + U2. Its value is equal to inverse of. Singe Of you. Divided by A Plus a constant of integration c. So in this case, our a value is 4 because +42 is equal to a 2, and we're going to get inverse of cinch. Of EU divided by 4. We want to evaluate the result from 0 to 4. And applying the fundamental theorem of calculus we're going to get inverse of sing of 4 divided by 4, which is 1. Minus Inverse of cinch. Of 0. What we can do from here is simplify inverse of sin of 0 is 0, so we basically get. Inverse of cinch of a 1. And finally We can express inverse of cinch in terms of the natural logarithm. We're going to remember that inverse of cinch of x is equal to ln of. X plus square root of X2 + 1 So in verse of cinch of one is going to be Ellen of. 1 + square root of 12 + 1, that's 2. And that's our final answer, Ellen of 1 + square root of 2, which corresponds to the answer choice D. Thank you for watching.

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.∫ dy / (y√(1 + (ln y)²)) from 1 to e

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In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ dy / (y√(1 + (ln y)²)) from 1 to e