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Numerade Educator

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Problem 10 Medium Difficulty

Find the general indefinite integral.

$ \displaystyle \int \sqrt{t} (t^2 + 3t + 2) \, dt $

Answer

$\frac{2}{7} t^{7 / 2}+\frac{6}{5} t^{5 / 2}+\frac{4}{3} t^{3 / 2}+C$

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Video Transcript

uh The theme in this problem is the power rule X. To the N. D. S. Which is adding one to the exponents and then dividing by your new expo or what I like to do is multiply by the reciprocal and don't forget about plus C. Because the derivative of a constant lazy river and the derivative of the thing and just wrote down needs to equal that. And this does, the only thing is make sure that this is a sum or difference of power. So as I'm looking at this, the square root of tee times, we can't do this rule until we simplify and we have T squared plus three T uh plus two. So my encouragement, as soon as first of all, don't write the square root, write it as T. To the one half hour. So then when you distribute in there, you know that you can just add your exponents. So I'm not doing the anti directive yet. Just rewriting as T. To the five house power Because that's 1/2 plus two. Uhh because when you multiply at the same base, you add the exponents and I ran out of space, but there should be A. D. T. Over here. So now I'm ready to do the anti griddle which is adding one to the exponential. Five house plus one is seven house. And that's why I don't like dividing by that new number because it looks prettier. If you multiply by the reciprocal, same thing with this T. To the five house power when you add one to it and when you multiply three By the reciprocal of two fists, you get six fists and the key to the three house power. And when you multiply two by the reciprocal of three half, so two times two thirds gives me four thirds, just like before you have to remember plus C. This is your correct answer, assuming I did all my math correct