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Point) Evaluate the indefinite integral:+84Answer:...

Question

Point) Evaluate the indefinite integral:+84Answer:

point) Evaluate the indefinite integral: +84 Answer:



Answers

Evaluate the given definite integral. $\int_{2}^{7} 4 d x$

Okay, so we're gonna be finding the value of the integral of three times dx. And so whenever we have a constant multiplied by um the function that we are taking, the integral of, we can actually take that constant outside of the integral and multiply it by just the integral of that function. So this is equal to three times the integral of dx. In the integral of dx. It's just gonna be equal to X. So this is gonna be three times X, Which is equal to three X. And then we also need to add our constant of integration. Since if we took the derivative of this function and we had a constant here, that constant would go to zero and so it would disappear when we take the derivative. And so that's why we always have these constants of integration that we need to add when we're finding um indefinite integral. And so the integral here of three D. X is equal to three X plus C.

Okay, so we're looking at this integral of X squared, then race to the one third power. Um due to properties of exponents. Whenever we have a number, a race to a power then raised to another power. This is going to be equal to that number. Race to those powers multiplied together. And so this is going to be equal to the integral of X to the two times one third power, which is two thirds power and then multiplied by dx. And so here we can just go ahead and take this integral and it's going to be involving a power rule. So if we look at the power rule for derivatives, we have the derivative of X. D. N is equal to end times X. D n minus one. So if we took the integral of both sides, this integral and derivative are going to cancel. So we're gonna have X. The N is equal to the integral of end times x the N -1. We can take this end out and divide both sides by N. So we get X. The end divided by N is equal to the integral Of X. The N -1. And I even put a DX here. And so this is the formula that I'm gonna be using to find this integral. And so I'll go ahead and move it just up here over here and then I'll erase this work that we did. And so here and -1 is going to be equal to 2/3. So we're gonna need to add one. So two thirds plus one would be two thirds plus three thirds or five thirds. We have extra five thirds. And then we divide by that power which is the same as multiplying by the reciprocal of that power. So that means we're gonna be multiplying by 3/5. And so this integral is equal to three times X. The five thirds divided by five. And then we need to add our constant of integration plus C.

Definite integral three D. X. From two, two full. So here evaluate the definite intrigue all. So here, first of all we have to pull out the constant. So here three times definitely and real D. X from two 24. So here we know that the integral of one is X. So we put here three times X. Two to fool. So now here we have to use the fundamental theorem of calculus. So here we get three times four negative two equal to six. So here we can see the value of definitely intrigued all three D. X. Oh, so it from 2 to 4 Is six, so it is over final answer.

Of one over to the bar so we can rewrite this to make it a little bit easier out of the integral of us. To the negative words. Yeah, on. Then, you can apply. Universe are able to take our time. So when we do that, we'll be out of one. Should be excellent. And divide that exploded. And so we add one. She made it for over three. We're gonna get this one. So 2 30 on. Then we divide by the excellent and divided by one of the three most. We can make it three accident. I get 1/3. Andi existed indefinitely. That plus C and you can rewrite this to get rid of our negative exploded as like it is free. Over. I keep brute. Oh, us. The keeper is a cover to the 1/3 power us. And this is the violence


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