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JH
Numerade Educator

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Problem 23 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int_0^1 (1 + \sqrt{x})^8\ dx $

Answer

$\frac{4097}{45}$

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Video Transcript

try and use substitution here. Let's take you to be one plus the square root of X, Then do you one over to radical X T X. Now, if we use our original u sub, we can rewrite rue X equals you minus one. So here we have two radical X equals two times you minus one So we can rewrite this most less multiplied the DX to the other side and related his two times you minus one and then on the right we just have the X. So this is what we'Ll were placed the ex within the new integral So we now have inaugural Also, let's watch out for these limits of integration. Originally, we have X equals zero Plug that into the use of so you have u equals one. Similarly, if x equals one, plug that in and you get you whose new equals two And then here we have one plus the radical so that you also they power. And then, as we saw a few moments ago, the X to you on this one. Pull out that too. And then here just used the power rule twice. So we have to you the ten over ten to you, the nine over nine and points or one into. Oh, so it's good. And plug those in. So we have. Well, first, you could cancel. Here. This is one over five, one over five to the tent, minus two or nine to the night. And then you plug in one. So one over five minus two over nine. And that their simplifying this we should get four o nine seven all over forty five. And that's your final answer.