Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit.

Consider the indefinite integral \displaystyle \int \frac{9 x^3+8 x^2+ 20 x + 16}{x^4+4 x^2}\, dx
Then the integrand has partial fractions decomposition

\frac{a}{x^2} + \frac{b}{x} + \frac{cx + d}{x^2+4}
where
a =
b =
c =
d =
Integrating term by term, we obtain that

After resolving into partial fractions,

a = 4

b = 5

c = 4

d = 4

The partial fraction obtained

(4/x²) + (5/x) + [(4x + 4)/(x² + 4)

On integration, we obtain

-(4/x) + 5 In |x| + 2 In |x² + 4| + 2 arctan (x/2) + C

where C is the constant of integration.

Step-by-step explanation:

(9x³ + 8x² + 20x + 16)/(x⁴ + 4x²)

This expression is rewritten as

(9x³ + 8x² + 20x + 16)/[x²(x² + 4x)]

This is then resolved into partial fractions

{(9x³ + 8x² + 20x + 16)/[x²(x² + 4x)]}

= (a/x²) + (b/x) + [(cx + d)/(x² + 4)

a, b, c and d are going to be obtained using partial fractions. This is presented in the attached image to this answer.

a = 4

b = 5

c = 4

d = 4

{(9x³ + 8x² + 20x + 16)/[x²(x² + 4x)]}

= (4/x²) + (5/x) + [(4x + 4)/(x² + 4)

= (4/x²) + (5/x) + [4x/(x² + 4)] + [4/(x² + 4)]

Integrating this,

∫ (4/x²) dx + ∫ (5/x) dx + ∫ [4x/(x² + 4)] dx + ∫ [4/(x² + 4)] dx

-(4/x) + 5 In |x| + 2 In |x² + 4| + 2 arctan (x/2) + C

where C is the constant of integration.

The integration is shown more properly on the page 2 of the attached image.

Hope this Helps!!!