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!!!

answer: e 9, 5.75

step-by-step explanation:

step 1

find the variation ab in the x coordinate

delta x=(x2-x1)=10-> 8 units

find the variation ab  in the y coordinate 

delta y=(y2-y1)=6-> 2 units

step 2

find the coordinates point c

  x coordinate 

cx=ax+delta x*2/> cx=2+8*(2/> cx=3

  y coordinate 

cy=ay+delta y*2/> cy=4+2*(2/> cy=4.25

the coordinates of point c are (3,4.25)

step 3

find the coordinates point d

  x  coordinate 

dx=ax+delta x*4/> dx=2+8*(4/> dx=4

  y  coordinate 

dy=ay+delta y*4/> dy=4+2*(4/> dy=4.5

the coordinates of point d are (4, 4.5)

step 4

find the coordinates point e

  x  coordinate 

ex=ax+delta x*4/> ex=2+8*(6/> ex=5

  y  coordinate 

ey=ay+delta y*4/> ey=4+2*(6/> ey=4.75

the coordinates of point e are (5, 4.75)

step 5

find the coordinates point f

  x  coordinate 

fx=ax+delta x*4/> fx=2+8*(8/> fx=6

  y  coordinate 

fy=ay+delta y*4/> fy=4+2*(8/> fy=5

the coordinates of point f are (6,5)

step 6

find the coordinates point g

  x  coordinate 

gx=ax+delta x*4/> gx=2+8*(10/> gx=7

  y  coordinate 

gy=ay+delta y*4/> gy=4+2*(10/> gy=5.25

the coordinates of point g are (7,5.25)

step 7

find the coordinates point h

  x  coordinate 

hx=ax+delta x*4/> hx=2+8*(12/> hx=8

  y  coordinate 

hy=ay+delta y*4/> hy=4+2*(12/> hy=5.5

the coordinates of point h are (8,5.5)

step 8

find the coordinates point i

  x  coordinate 

ix=ax+delta x*4/> ix=2+8*(14/> ix=9

  y  coordinate 

iy=ay+delta y*4/> iy=4+2*(14/> iy=5.75

the coordinates of point i are (9,5.75)

- r3ktforgood ☕

this is false ^.^ lol