Question

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

Answer 1

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:

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

$C\approx6.21$

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