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Find the most general antiderivative of the function (Check our answer by differentiation Use € for the constant of the antiderivative:) f(x) = (x + 3)(Sx 10)...

Question

Find the most general antiderivative of the function (Check our answer by differentiation Use € for the constant of the antiderivative:) f(x) = (x + 3)(Sx 10)F(x)

Find the most general antiderivative of the function (Check our answer by differentiation Use € for the constant of the antiderivative:) f(x) = (x + 3)(Sx 10) F(x)



Answers

Find the most general antiderivative of the function.(Check your answer by differentiation.)
$f(x)=\frac{1+x-x^{2}}{x}$

Okay, so for this question, we have f of X is equal to e square. So if we take the anti derivative e squared is just a conscience that just gets multiplied by acts. Plus c, we take the derivatives. The street kind of acts is just e squared.

Were given a falling function. Ever have access he could accept that might accept their plus two X All were extra. 14 were asked on inter derivative of this function. We were asked to check her work using differentiation. So shut off with Let's bring X 1/4 up genuine writer. So I looked with X minus exit native one plus two x 2 93 And so then we can use a reverse power role to find our derivatives. So we have our trainer Inter derivative. So we have 1/2 x squared and then we know that the interpretive off accident of one inches national log of the absolute value of X Then we have to accidentally three. So they bump up trucks and entry to trucks amended to and then we divide to buy chew By night of choose, we get minus X who didn't go to? If we can then right? I was X squared over two Mr Natural log of severely vax There's one over x squared. We're also can not forget our constant whoever with on anti terror of this so that we can differentiate this entire thing we get X my ass vaccinated one plus two x in 93 plus zero when you see that this much is up with original function. So therefore we know that this is anti drifted.

Suppose you have F of X. Which is equal to to raise to expose for hyperbolic sine of X. I mean here we want to find the most general anti derivative of this function. So boys be led F of X as the anti derivative of the function. Now note that the derivative of to raise two X. This is equal to to raise two X times L n F two. And so if you take the derivative of to race to X over Ellen of two, we get two race to X Times Ln of two. This all over Ellen of two. We have to raise two x. So for to raise two X. Our anti derivative would be to raise two X over L N. F. Two. Now for the four times hyperbolic sine of x. We know that the derivative of the hyperbolic cosine of X. This is equal to hyperbolic sine of X. And so if we take the derivative of the Function four times the hyperbolic co sign of X. This will give us four times the hyperbolic sine of x. And so from here we know that the Anti derivative of four times the hyperbolic sine of x. As equal to four times the hyperbolic co sign off X. And so combining these. Now we have F. Of X. Which is the anti derivative of F of X. This is equal to to race to X Over l. n. f. two Plus we have four times the hyperbolic co sign of X. And because you want the general anti derivative we have to add constant C. Were seized any arbitrary real number. Now to check if we have the correct and derivative, we will take the derivative of F of X. Now, since the derivative of F of X, this is equal to derivative of To raise two x over LNF two Plus four co sign H. X. Policy. This is just we have one over LNF two times to race two X Times LNF two Plus We have four times hyperbolic sine of X plus zero. This gives us to raise two X plus four times the hyperbolic sine of x. And so you have the correct and derivative.

This question strikes us to find the anti derivative. The first thing we know is that we have experts one times two X minus one. I would highly recommend distributing this and writing it and its simplest form. This is gonna make it a lot easier to now Take the integral now to take the integral we know we increase the exponents I won and then make the exponents multiplied by the fraction what the original coefficient waas. So same for this term, we increased the exponents to from one make this 1/2 and then this is minus one axe pussy and then we know this is correct. If you take the derivative, we get three times to over three, which is two x squared plus x. Just exit the first power minus one.


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