stuck on partial fraction integration problem?

Question

integrate : (s^4 + 81) / s(s^2 +9)^2 ds

u = s^2 + 9
du = 2 sds
sds = 1/2 du

came up with

A/s + (Bs +C/ s^2 + 9) + (Ds + E / (S^2 +9)^2)


What do I need to do next?

Answer 1

I haven't done this in ages, but I think I would hold off on defining u until later and multiply out the term in the denominator. Then you have

(s^4+81)/ [s(s^4+81)+18s^3]
maybe that a better jumping off point into the partial fraction.

Answer 2

it is going to be: (-19) / ((2x + 9)(3x + 4)) = ((A) / (2x + 9)) + ((B) / (3x + 4)) simple denominator (-19) / ((2x + 9)(3x + 4)) = (A(3x + 4)) + (B(2x + 9)) / ((2x + 9)(3x + 4)) on condition that we've the comparable denominator we can drop them -19 = A(3x + 4) + B(2x + 9) clean up for A by using plugging in -4.5 -19 = A(-13.5+ 4) + B(-9 + 9) -19 = -9.5A A = 2 clean up for B by using plugging in -4/3 for x -19 = A(-4 + 4) + B(-8/3 + 9) -19 = B(19/3) B = -3 (-19) / ((2x + 9)(3x + 4)) = ((2) / (2x + 9)) - ((3) / (3x + 4)) combine the appropriate factor ln(2x + 9) - ln(3x + 4) i'm hoping this permits. Have a solid day.