Problem 74 Hard Difficulty

If $ f $ is a quadratic function such that $ f(0) = 1 $ and
$$ \int \frac{f(x)}{x^2 (x + 1)^3}\ dx $$
is a rational function, find the value of $ f^\prime (0) $.

Answer

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Video Transcript

F is a quadratic function. F zero was one and were given that this in a girl's irrational function and would like to find at prime of zero. So let's come back up here and use this information. We have zero equal C equals one. So that gives us see So we have a of X equals x, where, plus b X plus one. And the thing that we're after is a primal zero. So that's just need. So we'd like to find the value of be such that this thing is rational function, so that will determine the possible values for me. So let's go ahead and do partial fraction to composition. So right now our instagram is f of X, which we know is that up there and then we have X squared. That's plus one cute. Now, using what the author calls case, too, I can write. This is ale rex be over ex players and then it using case to again, this time for X plus one she over X plus one deal Rex plus one square e over at plus one. Cute Oops And then let's go ahead and most supply Well, actually, at this case was before we multiply, we may not have to hear. Let's just note that we want this intervals to be irrational function. So, in particular, rational means we don't want logarithms No lots here, so that automatically eliminates these two terms here. We do not want these terms because these will give us natural logs. So this gives us a and sear both zero. So when I go ahead and multiply both sides by this denominator and we're getting rid of A and C score the next page, I get a X Square, be explicit one and then on the right, and then we could go ahead and solve this. So go ahead and expand the sow and then combine based on the power of ex supporter X cubed pullout of X squared. And that's what we're doing right now. So that's when you pull out X Square and then three b x. Well, then the and then we end up with three. B equals little bee. Large B is one, so that gives us B equals three. And that's exactly what we wanted. Because remember from the previous page, we wanted a crime of zero. We knew that was deep the be over here in front of the X, and we just found that value will be that gives us three, and that's your final answer.