indefinite integrals

Question

My daughter is stumped on this. She was only able to get part of the problem correct. Any help is appreciated.

Consider the indefinite integral (7x^3+7x^2+20x+16)/(x^4+4x^2)dx

Then the integrand has particular fractions decomposition

(a/x^2)+(b/x)+(cx+d/x^2+4)

where

a=?

b=?

c=2

d=?

Integrating term by term, we obtain that integral (7x^3+7x^2+20x+16)/(x^4+4x^2)dx =?

She has to solve for every spot there is a "?". Any help on showing the steps is greatly appreciated.

Answer 1

Answer 2

(7x^3+7x^2+20x+16)/(x^4+4x^2) = ( A/x²) + (B/x) +(C/(x²+4) + (D/(x⁴+4x²), Four terms needed to match left side.

Multiply both sides by x⁴ + 4x²

7x^3+7x^2+20x+16 = A(x²+ 4) + Bx(x²+ 4) + Cx² + D

7x^3+7x^2+20x+16 = Ax² + 4A + Bx³ + 4Bx + Cx² + D

7x^3+7x^2+20x+16 = Ax² + Cx² + Bx³ + 4Bx + D

7x^3+7x^2+20x+16 = Bx³ + (A + C)x² + 4Bx + D

D = 1 6

B = 7

A + C = 7

4B = 20

Answer 3

integral = 5ln|x|+ ln(x^2+4)- 4/x +3arctan(x/2)/2 + C

(2x+3)/(x^2+4) + 5/x + 4/x^2

a=4

b=5

c=2

d=3