Answer to Question #110462 in Calculus for Flavia

Question

Integrate x^2+X+5/(x^2+4)(X+1) dx

Answer 1

$∫(x^2+x+5)/((x^2+4)(x+1)) dx=⊗$

$(x^2+x+5)/((x^2+4)(x+1))=(Ax+B)/(x^2+4)+C/(x+1)$ ;

$(Ax+B)(x+1)+C(x^2+4)=x^2+x+5;$

$x^2:A+C=1;$

$x:A+B=1;$

$x^0:B+4C=5;$

$C=1-A;B=1-A;1-A+4(1-A)=5;-5A=0;$

$A=0;B=1;C=1;$

$⊗=∫dx/(x^2+4)+ ∫dx/(x+1)= (1/2) arctan (x/2)+ln⁡|x+1|+C.$