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JH
Numerade Educator

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Problem 46 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{(x - 1) e^x}{x^2}\ dx $

Answer

$\frac{e^{x}}{x}+c$

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Video Transcript

Let's use integration by parts here. Let's take you to be the numerator or, if you like, you could rewrite this and then this implies, Do you? So here on the first term, use the product rule and then for the last one, that's just you to the ex. Good. Cancel those first, the last terms and we see that we just have x e x, the ex. So that's hard to you. And then Devi is whatever's left over in the inside Grant that'LL be extra the minus two DX and then go ahead and use the power rule there to get negative one over X. So now we're ready to use the formula Rickel U V minus integral d to you. Now it's good and plug in. Are you an RV? So go ahead and plug in and simplify. So that's you times V and then minus. And then here we have our liberal and let's do we have V and then we also have do you as well. So here's our B and we have our negative one of Rex the X as well. So now let's go ahead. So again, this is coming from the DX Times X to the X is coming corresponding to these two terms here. So the last step here. So just before we integrate cross off thes exes and then go ahead and multiply these two negatives together to get a plus, Let me go into the next page here. Thank you. So we had minus E x X minus one over X thousand, the UV. And then after cancelling those double negatives, we just have integrally X. So here we could even go ahead and multiply this out. So that's the first term up top, and then we have plus eating eggs. These are both over X. And then here we have plus E to the X plus he that cancel out those exes and then we have a minus e X, and that cancels with the plus X. So we're just left over with either the ex over X plus, see? And that's our final answer.