Evaluate each integral.

$$

\int_{0}^{5}\left(x^{4} y+y\right) d x

$$

630$y$

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All right, So we're going to the Inter Girl. Um, basically the integral from 0 to 5 of the function x the fourth times y plus why t x and I know what you're thinking. It's a little confusing efforts because you see two variables, but there's no, like double integral. So don't be worried because, um, you're not trying to find a numerical answer in this poem, but more of a you're gonna have variables in your answering them, so don't worry about it too much. So let's start. Um, so I'm just gonna factor out the why on this is because we have a D accident roll, meaning we're integrating respect, Teoh X. So every other variable is treated as a constant. So by factoring out this, why I can then move this wide to the outside of the integral, as if they were a constant, because we are integrators by two. Thanks. Okay, so now I have this integral. And now what we can dio is just used the reverse of the power rule in order to integrate Excellent fourth. So we just add one to the exponent and then divide by that new exponents. So excellent fifth Divide by five and then plus X. And we want to evaluate this from 0 to 5. So just playing in, um, we get five to the fifth over five, plus five, and then we want to subtract the second component, Um, which should just be a zero plus zero. Okay, It's gonna be equal to just gonna simplify that. Five to the fourth, because five to the fifth of I 55 to the fourth plus five we have. Why times five to the fourth is 6 25 on at five. So our final answer should be 6 30 times. Why?

Rutgers, The State University of New Jersey