Answer to Question #330783 in Calculus for Quân Jason

Question #330783

F(x) = "\\smallint" "(x+9)dx \/ (x-2)(x+2)"2 and F(3.1) = 5.5. Compute F( -0.5)


1
Expert's answer
2022-04-19T15:31:14-0400

Decomposition into the partial fraction:

"\\frac {x+9}{(x-2)(x+2)^2}=\\frac {A}{x-2}+\\frac{B}{x+2}+\\frac{C}{(x+2)^2}"

"x+9=A(x+2)^2+B(x-2)(x+2)+C(x-2)"

Equating coefficients, we get

"A=\\frac{11}{16}" ; "B=-\\frac{11}{16}" ; "C=-\\frac{7}{4}" .

"F(x)=\\int \\frac {x+9}{(x-2)(x+2)^2}dx=""\\int (\\frac {11}{16}(\\frac {1}{x-2}-\\frac{1}{x+2})-\\frac 74\\frac{1}{(x+2)^2})dx=""\\frac{11}{16}\\ln {|\\frac{x-2}{x+2}|}+\\frac74\\frac {1}{x+2}+C"

Let’s find C:

"F(3.1)= \\frac{11}{16}\\ln {|\\frac{3.1-2}{3.1+2}|}+\\frac74\\frac {1}{3.1+2}+C=5.5"

"C\\approx6.21"

"F(-0.5)= \\frac{11}{16}\\ln {|\\frac{-0.5-2}{-0.5+2}|}+\\frac74\\frac {1}{-0.5+2}+6.21\\approx7.73"



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