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Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.$$\int \frac{e^{\sqrt{x-1}}}{\sqrt{x-1}} d x$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 1

An Overview of Integration Methods

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

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Evaluate the integrals.

So in this question, we're going to be solving the indefinite integral. Using the method of the substitution, we have the integral of E to the square root of X minus one over the square root of X minus one. So first thing we're going to do is we're going to set. Are you equal to the square root of X minus one? And I we're gonna take the derivative of both sides. And so the derivative of the square root of X minus one. What we're going to do for that, just as a reminder is, we're going to take both. We're going to the powerful and the chain room. So this is going to be equal to X minus one to the one half, and from here, we're going tohave one half times X minus one. Uh, yes. Times X minus one to the one half minus one. And this is going to be times our internal derivative of just one. So we're now going tohave one half times X minus one to the negative one half, which is just going to be equal to 1/2 times the square root of X minus one. De looks I'll put a d x here. Just a reminder. And so from here, what we're going to do is we're going to replace R square root of the X minus one with argue and r d x times X minus one with r D r two d u Because we're gonna bring are gonna multiply our two times are D'You um, and times both sides really to get it out of our denominator here and now we can be right our equation. So we're just going to have the integral of e to the square root. I have, uh, not square root. Sorry. Each of the U I am two d u times to do you. I am so from here we can take out Are you? I was just going to have the integral of e to the U D you and the integral of each of the u. D. You is just equal to each of the you, so we're going to have to times e to the U plus r c on from here. We have to just fill in for our you and are you is equal to the square root of X minus one right And so we're going to just have to times e to the square root of X minus one. Let's see, and that is our final answer.

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