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!!!

idk

step-by-step explanation:

the correct option is 2.

step-by-step explanation:

according to the given information triangle abc is a right angle triangle with right angle at a. segment ab is 5 and segment ac is 7. point d is on segment bc and angles adb and adc are right angles.

it is given that triangles abd, cad, and cba are similar.

two triangles are similar if their corresponding sides are proportional.

in triangle cba, using pythagoras theorem

                      )

triangle abd and cba are similar,

                            (2)

triangle cad and cba are similar,

                      )

using (1), (2) and (3), we get

hence proved.

therefore option 2 is correct.