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