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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___________

Answer :

Answer:

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!!!

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