use the method of equating coefficients.
integral of (x^3+5x^2+12)/((x^2)(x^2+4)) dx
(x^3+5x^2+12)/((x^2)(x^2+4)) = (x+2)/(x^2+4) + 3/x^2
So,
∫(x^3+5x^2+12)/((x^2)(x^2+4))
= ∫ (x+2)/(x^2+4) + 3/x^2 dx
= 1/2 log(x^2+4) + arctan(x/2) - 3/x + C
equating coefficients = partial fractions ?
let (x^3+5x^2+12)/((x^2)(x^2+4))
= A/x^2 + (Bx+C)/(x^2+)
= (Ax^2 + 4A + Bx^3 + Cx^2)/(x^2(x^2+4))
= (Bx^3 + (A+C)x^2 + 4A)/(x^2(x^2+4))
so B=1
A+C=5
4A=12 ---> A = 3
in A+C=5 ---> C=2
so we can write:
(x^3+5x^2+12)/((x^2)(x^2+4))
=3/x^2 +(x+2)/(x^2+4)
and ∫(x^3+5x^2+12)/((x^2)(x^2+4)) dx
= ∫ 3/x^2 dx + ∫ (x+2)/(x^2+4)dx
= -3/x + (1/2)ln(x^2+4) + PPPPP
where PPPP = ∫ 4/(x^2+4) dx
from my formula tables from "years ago" this is
tan^-1 (x/2)
= -3/x + (1/2)ln(x^2+4) + tan^-1 (x/2) + constant
better check this, I did not write it out on paper first.
To find the integral of the given function using the method of equating coefficients, follow these steps:
Step 1: First, rewrite the given function as partial fractions. To do this, factor the denominator:
x^2(x^2 + 4) = (x^2)(x + 2i)(x - 2i)
Step 2: Express the given function as a sum of partial fractions:
(x^3 + 5x^2 + 12) / (x^2)(x^2 + 4) = A/x^2 + (Bx + C)/(x + 2i) + (Dx + E)/(x - 2i)
Step 3: Multiply the equation by the common denominator (x^2)(x^2 + 4) to eliminate the denominator:
x^3 + 5x^2 + 12 = A(x + 2i)(x - 2i) + (Bx + C)(x^2)(x - 2i) + (Dx + E)(x^2 + 4)
Step 4: Expand and simplify the equation:
x^3 + 5x^2 + 12 = A(x^2 - (2i)^2) + (Bx + C)(x^3 - 2ix^2) + (Dx + E)(x^2 + 4)
Step 5: Equate the coefficients of like terms on both sides of the equation. This will allow us to solve for the unknown coefficients {A, B, C, D, E}:
Coefficient of x^3: 0 = B + D
Coefficient of x^2: 1 = A - 2B + D + E
Coefficient of x^1: 0 = C - 4E
Coefficient of x^0: 5 = -4A + 4C + 4E
Step 6: Solve the system of equations for the coefficients. This can be done by substitution or elimination, depending on your preference. Once you solve for the coefficients, you will have the partial fraction decomposition.
Step 7: Now, integrate each partial fraction separately. The integral of 1/x^2 is -1/x, and you can apply power rule for the other terms. For example, the integral of (Bx + C)/(x + 2i) can be found by substitution, setting u = x + 2i. After evaluating the integrals, you will have the final result.