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This question is quite tricky. It's for my Calculus 2 assignment and I can't seem to figure out how to integrate this function in order to get its derivative. I tried partial fractions, u-sub with x-5 and x^2, but nothing seems to work. All I conluded is that the quadratic function y-coordinate is -4.enter image description here

Can someone help me with this?

Regards, You Xiao Ruan.

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    $\begingroup$ The numerator has to be of the form $ax^2+bx-4$ and when you integrate given integral, there cannot be an $ln$. This means that somehow the division has to come out "evenly",i.e. stuff has to cancel out. Now the denominator has two zeros... $\endgroup$
    – imranfat
    Commented Apr 2, 2017 at 2:53
  • $\begingroup$ Would a u-sub be helpful here? $\endgroup$
    – You Xiao Ruan
    Commented Apr 2, 2017 at 2:57
  • $\begingroup$ I don't think so. But the zeros of the denominator are very helpful to determine $a$ and $b$. Then stuff is supposed to cancel $\endgroup$
    – imranfat
    Commented Apr 2, 2017 at 3:14
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    $\begingroup$ You want $ax^2+bx-4$ to be divisible by (x-5) for sure. Maybe by $(x-5)^2$....what do you think? $\endgroup$
    – imranfat
    Commented Apr 2, 2017 at 3:18
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    $\begingroup$ Actually I perceive a little problem here with the exercise. The thing is that in order not to have an $ln$ as part of an anti derivative, you cannot have a linear term after partial fraction decomposition. I was at first thinking of $f(x)=-4/25(x-5)^2$ because that passes through $(0,-4)$, but after canceling, I checked and it ends up with an $ln$ after integrating. So this can't be it. There has to be some other "non-conventional" trick, which I don't see at the moment. It is getting late here and half my brain is asleep...I did upvote your question. $\endgroup$
    – imranfat
    Commented Apr 2, 2017 at 3:36

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