Evaluate each integral.

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

$=630 y$

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Johns Hopkins University

Harvey Mudd College

University of Nottingham

Idaho State University

in this problem here were asked, defying the integral of a constant times integral from 0 to 5 of the quantity exit fourth wife Plus Why DX Now the key here is with respect X. That means that as we're doing this in a role, any variable that isn't X we can treat is a constant at that point that will make this fairly logical. So let's start by taking our constant. We were just factored out front. And now let's do the anti derivative of these terms. Well, if why is constant well, then the integral of extra forthis X to the fifth over, five times that constant? Why, plus the second term again? If we're integrating with respect to X, why is constant then? The integral of why is just x times that constant and then you read Evaluate. This integral from X equals five X equals zero, and then that will be multiplied by C. Well, if we use the fundamental theory, my calculus and plug in five for X and then subtract groups and that's a tract plugging in zero franks, well, that's a museum up zero. So on the lower than the integration, both those are zero. Well, now we can sit by this little bit. Five for this fifth over five. This stain was 5 4625 Why? Plus five? Why all of that times this constant C and that all together makes 630. Why? Times the constancy. The important part here was in with respect to x the integral any of these variables inside our Constance. Who says if this was the number seven or the number 14 when the variables different, treated as a constant.

University of Nebraska - Lincoln