cherrilyn
  • cherrilyn
evaluate the integral
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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cherrilyn
  • cherrilyn
\[\int\limits_{?}^{?}(x ^{2} + 3x -44)dx/(x+3)(x+5)(3x-2)\]
amistre64
  • amistre64
partial fraction decomp it looks like; unless we can factor the top to cancel some bottoms
amistre64
  • amistre64
x^2 +3x -44 = A(x+5)(3x-2) ; x = -3 x^2 +3x -44 = B(x+3)(3x-2) ; x = -5 x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3

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cherrilyn
  • cherrilyn
woohoo thats what I got. but now how do I find the actual values of the letters?
amistre64
  • amistre64
-44 = A(2)(-11) -44/-22=A A = 2 ---------------------- 25 -59 = B(-2)(-17) -34 = B(34) B = -1 x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3
amistre64
  • amistre64
when x = a number that zeros the other 2; use it and solve for that letter
cherrilyn
  • cherrilyn
but sometimes it leaves behind more than one variable
amistre64
  • amistre64
x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3 (2/3)^2 +3(2/3) - 44 = C(2/3 +3)(2/3 +5)
amistre64
  • amistre64
sometimes it does, but use it first to get some solids :) with linears on the bottom, its not a problem
cherrilyn
  • cherrilyn
okay. for the last equation I got 4/9 - 42 = c(2)(2/3 + 5)
amistre64
  • amistre64
(4/9) +2 - 44 = C(11/3)(17/3) (4/9) - 42 = C(11/3)(17/3)
amistre64
  • amistre64
-374/9 = C(187/9) C = -374/189 reduce as wanted
cherrilyn
  • cherrilyn
wow how do you work so fast?
amistre64
  • amistre64
i use my head and mess up alot lol
cherrilyn
  • cherrilyn
lol oh... do you think you can help me with another similar problem?
amistre64
  • amistre64
perhaps; but let me get thru this one first :)
cherrilyn
  • cherrilyn
okay! thanks for all of your help :)
amistre64
  • amistre64
we did C correctly right? that monster fraction scares me into thinking I messed it up lol
amistre64
  • amistre64
i typoed the 187 into a 189 ;)
amistre64
  • amistre64
C = 2 :)
amistre64
  • amistre64
-2
cherrilyn
  • cherrilyn
wait, in this problem.. there is no Dx+E?
amistre64
  • amistre64
so our integral becomes: [S] 2/(x+3) - 1/(x+5) - 2/(3x-2) dx
amistre64
  • amistre64
the bottom is already factored for us into linear forms; linear form dont need the Dx+E form
cherrilyn
  • cherrilyn
ahh okay! Yayyyy. :D
amistre64
  • amistre64
if we had an irreducible quadratic in the denom; then we would use the Dx+E form above it to aid us
cherrilyn
  • cherrilyn
like.. (x-1)^3?
amistre64
  • amistre64
nope; that is a multiple of a linear and still only needs a A B C single atop it
cherrilyn
  • cherrilyn
really? how would it look.. A/x-1 + B/x-1 + c/x-1?
amistre64
  • amistre64
x^2 + 10 is irreducible quadratic; cant be factored to real linears
amistre64
  • amistre64
A/(x-1) + B/(x-1)^2 + C/(x-1)^3
cherrilyn
  • cherrilyn
ohh.. hmm . and that would be... Ax+B/x^2+10?
amistre64
  • amistre64
our integral has become: 2ln(x+3) - ln(x+5) -2ln(3x-2) +C
cherrilyn
  • cherrilyn
thats the final answer
amistre64
  • amistre64
thats the form we would use above it yes :)
amistre64
  • amistre64
we could probably turn this into one ln(yadayada)+C if we want, but this is good :)
cherrilyn
  • cherrilyn
wow thank you!!!!
amistre64
  • amistre64
youre welcome :)

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