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JH
Numerade Educator

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Problem 28 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \sin \sqrt{at}\ dt $

Answer

$\int \sin (\sqrt{a t}) d t=\frac{2}{a}\{-(\sqrt{a t}) \cos (\sqrt{a t})+\sin (\sqrt{a t})+C\}$

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Video Transcript

Let's start off here by doing you use up that here, Do you? Using the chain rule from capitals. We could rewrite this as a over to you DT. And let's go ahead and solve this equation for DT. So we have dt equals to you over a to you and that's whole replace over here in the original integral sign you replaces the first term and then here for DT We can use this so we have a new integral and let's pull out that constant. So on the downside, we do have this extra factor in front of the sign that we didn't have in the original. But the expression inside the sign is now much easier to the OS. So for this in a girl here, you can do integration. My parts Let's go ahead and take w to be you t w equals Do you t v equals sign you do you the equals negative co sign you and then you have to over a recall the formula for integration by parts You've even this case w v minus inner growth e d w. So using that formula here, there's a W times me and then in a girl Z and then DW So this is just negative to you, coz I knew over, eh? Plus two over, eh? Sign you And that was good. And add that sea and finally and the last step here is suggest go to your original substitution and took plug in okay with to plug in U equals radical eighty season. So that's the first term and then for the second term, and that's your final answer.