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Evaluate each integral.

$$

\int_{4}^{5} x \sqrt{x^{2}+3 y} d y

$$

$\left[\left(x^{2}+15\right)^{3 / 2}-\left(x^{2}+12\right)^{3 / 2}\right]$

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Johns Hopkins University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

{'transcript': "the integral of ex route X squared plus three. Why d y from 4 to 5? Uh, note that the d Y here means that we can treat all other variables as constants. So this X first of all, can come out of the integral. And we'll rewrite X squared plus three y to the power of 1/2. Using our exponents laws, the anti derivative for X squared plus three y to the power of 1/2 is to over three times x squared plus three y to the power of three over to make sure you account for the chain rule with the 1/3 at the end. And we evaluate this from 4 to 5 well below the constants which will make this to X over nine times. Ah, the substitution of the Y values. So X squared plus 15 to the power of three over two, minus x squared. Plus 12th to the power of three over two. Yeah, I don't know. Does that sound okay?"}