1. (integral sign) 2x+5(over) x(squared)+3x+6
2. (integral sign) x(cubed)+5x(squared)+4(over) x(x(squared)+3x+2)
****over is equivalent to the division sign

these are the answers we got, can you confirm if they are correct. When I put x=0 in question 2, LHS and RHS worked out to be the same. But just want some confirmation.
∫ (2x + 5)/(x² + 3x + 6) dx
= ∫ (2x + 3 + 2)/(x² + 3x + 6) dx
= ∫ (2x + 3)/(x² + 3x + 6) dx + 2 ∫ dx/(x² + 3x + 6)
= ∫ (2x + 3)/(x² + 3x + 6) dx + 2 ∫ dx/((x + 3/2)² + 15/4)
= ∫ (2x + 3)/(x² + 3x + 6) dx + 2 ∫ dx/((x + 3/2)² + √(15)/2)
= ln (x² + 3x + 6) + 2 (2/√15) arctan( 2(x + 3/2)/√15) + C
= ln (x² + 3x + 6) + (4/√15) arctan( 2(x + 3/2)/√15) + C

∫ (x³ + 5x² + 4)/[x(x² + 3x + 2) ] dx
= ∫ (x³ + 5x² + 4)/(x³ + 3x² + 2x) dx
= ∫ (x³ + 3x² + 2x + 2x² - 2x + 4)/(x³ + 3x² + 2x) dx
= ∫ (x³ + 3x² + 2x)/(x³ + 3x² + 2x) dx + ∫ (2x² - 2x + 4)/(x³ + 3x² + 2x) dx

Let
(2x² - 2x + 4)/(x³ + 3x² + 2x) = A/x + B/(x + 1) + C/(x + 2)
Then
2x² - 2x + 4 = A(x + 1)(x + 2) + Bx(x + 2) + Cx(x + 1)
2x² - 2x + 4 = x²(A + B + C) + x(3A + 2B + C) + (2A)

By comparing coefficients of different powers of x:
2A = 4 ==> A = 2

-2 = 3A + 2B + C ==> -2 = 6 + 2B + C ==> 2B + C = -8

2 = A + B + C ==> 2 = 2 + B + C ==> B + C = 0 ==> B = -C

==> -2C + C = -8 ==> C = 8 ==> B = 8

==> (2x² - 2x + 4)/(x³ + 3x² + 2x) = 2/x + 8/(x + 1) - 8/(x + 2)

continuation
= ∫ (x³ + 3x² + 2x)/(x³ + 3x² + 2x) dx + ∫ (2x² - 2x + 4)/(x³ + 3x² + 2x) dx
= ∫ 1 dx + ∫ (2/x + 8/(x + 1) - 8/(x + 2)) dx
= x + 2ln|x| + 8ln|x + 1| - 8ln|x + 2| + C

Please, do not double post. Thank you.

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THREAD CLOSED

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