Expand:
(x^2 - 2)/((x + 1)(x^2 + 3)) = A/(x + 1) + (Bx + C)/(x^2 + 3)x2−2(x+1)(x2+3)=Ax+1+Bx+Cx2+3
x^2 - 2 = A(x^2 + 3) + (Bx + C)(x + 1)x2−2=A(x2+3)+(Bx+C)(x+1)
Let x = -1 to make B and C disappear:
-1^2 - 2 = A(-1^2 + 3)−12−2=A(−12+3)
-1 = 4A−1=4A
A = -1/4A=−14
x^2 - 2 = -1/4(x^2 + 3) + (Bx + C)(x + 1)x2−2=−14(x2+3)+(Bx+C)(x+1)
Let x = 0 to make B disappear:
- 2 = -1/4(3) + (C)(1)−2=−14(3)+(C)(1)
C = -5/4C=−54
x^2 - 2 = -1/4(x^2 + 3) + (Bx -5/4)(x + 1)x2−2=−14(x2+3)+(Bx−54)(x+1)
Let x = 1:
1^2 - 2 = -1/4(1^2 + 3) + (B -5/4)(1 + 1)12−2=−14(12+3)+(B−54)(1+1)
-1 = -1 + (2B -5/2)−1=−1+(2B−52)
B = 5/4B=54
int(x^2 - 2)/((x + 1)(x^2 + 3))dx = -1/4int1/(x + 1)dx + 5/4intx/(x^2 + 3)dx - 5/4int1/(x^2 + 3)dx∫x2−2(x+1)(x2+3)dx=−14∫1x+1dx+54∫xx2+3dx−54∫1x2+3dx
int(x^2 - 2)/((x + 1)(x^2 + 3))dx = -1/4ln|x + 1| + 5/4ln|x^2 + 3| - (5sqrt(3))/12tan^-1(x/sqrt(3))∫x2−2(x+1)(x2+3)dx=−14ln|x+1|+54ln∣∣x2+3∣∣−5√312tan−1(x√3)
