integral

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Find dy/dx and d2y/dx2 if y= definite integral sign where a= 1 and b= 3x
1/(t^2+t+1) dt

how do i even start. do i integrate and then plug in a and b? plz help.

  • integral -

    dy/dx = 1/(x^2+x+1) * 3

  • integral -

    can you explain how you got this. thanks

  • integral -

    excuse me? Didn't we just go through this about differentiating under the integral sign?

  • integral -

    but i don't get how you can just replace the variable t with x so easily. i don't get how you did that. and how does the 3 come into the picture for dy/dx.

  • integral -

    In general (see the wikipedia article for proof),

    ∫[a(x),b(x)] f(t) dt
    = f(b(x)) db/dx - f(a(x)) da/dx

    Since we have a(x) = 1, da/dx = 0 and we are left with

    ∫[a(x),b(x)] f(t) dt
    = f(b(x)) db/dx

    b(x) = 3x, so db/dx = 3

    So, oops. I made a mistake. It should be

    dy/dx = 1/((3x)^2+3x+1) * 3

    Good catch! :-)

  • integral -

    Thank you very much! I really didn't understand the form but now I do. :)

    Also, would d2y/dx2 be:

    -9(6x+1)/((9x^2+3x+1)^2)

    I just want to make sure that d2y/dx2 is asking for the second derivative?

  • integral -

    haven't checked your work, but yes, they want y"

  • integral -

    Oh okay thank you!

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