Math question....Integral?

Divide the polynomials by the comparing coefficients method:

(x² + 4x + 8) / (x + 1) = Ax + B + C / (x + 1)

x² + 4x + 8 = Ax(x + 1) + B(x + 1) + C

x² + 4x + 8 = Ax² + Ax + Bx + B + C

x² + 4x + 8 = Ax² + (A + B)x + (B + C)

A = 1

A + B = 4

B = 4 - A

B = 4 - 1

B = 3

B + C = 8

C = 8 - B

C = 8 - 3

C = 5

(x² + 4x + 8) / (x + 1) = x + 3 + 5 / (x + 1)

Integrate the expression term by term using this result:

∫ (x² + 4x + 8) / (x + 1) dx = ∫ [x + 3 + 5 / (x + 1)] dx

∫ (x² + 4x + 8) / (x + 1) dx = ∫ x dx + 3 ∫ 1 dx + 5 ∫ 1 / (x + 1) dx

∫ (x² + 4x + 8) / (x + 1) dx = x² / 2 + 3x + 5ln|x + 1| + C

(x^2 + 4x + 8) / (x + 1) = x + 3 + [5 / (x + 1) ]

The integration of this function is

[(x^2) / 2] + 3x + 5 log|x + 1| + C

(x^2 + 4x + 8) / (x + 1)

= (x^2 + x + 3x + 3 + 5) / (x + 1)

= x + 3 + 5/(x+1)

So, the integral = (1/2)x^2 + 3x + 5ln|x+1| + c

We divide x²+4x+8 through x+1, this gives

x+3 and 5/(x+1) as remainder

so we integrate x+3 = x²/2+3x+C

and 5/(x+1) = 5 ln( | 1+x | ) + C '

so add both we have

x²/2 + 3x + 5 ln (|1+x|) + C"

u = x + 1

du = dx

x = u - 1

(x^2 + 4x + 8) * dx / (x + 1) =>

((u - 1)^2 + 4 * (u - 1) + 8) * du / u =>

(u^2 - 2u + 1 + 4u - 4 + 8) * du / u =>

(u^2 + 2u + 5) * du / u =>

u * du + 2 * du + 5 * du/u

Integrate:

(1/2) * u^2 + 2u + 5 * ln|u| + C =>

(1/2) * (x + 1)^2 + 2 * (x + 1) + 5 * ln|x + 1| + C =>

(1/2) * (x^2 + 2x + 1) + 2x + 2 + 5 * ln|x + 1| + C =>

(1/2) * x^2 + x + (1/2) + 2x + 2 + 5 * ln|x + 1| + C =>

(1/2) * x^2 + 3x + 5 * ln|x + 1| + (5/2) + C =>

(1/2) * x^2 + 3x + 5 * ln|x + 1| + C =>

(1/2) * (x^2 + 6x + 10 * ln|x + 1|) + C

∫(x² + 4x + 8)/(x + 1) dx

∫((x + 2)² + 4)/(x + 1) dx

∫((x + 1)² + 2(x + 1) + 5)/(x + 1) dx

∫(x + 1) dx + ∫2 dx + ∫5/(x + 1) dx

= x²/2 + 3x + 5*ln|x + 1| + C

Eh... neither... yet i guess i will choose English. I used to love English extra yet extreme college made it boring. -_- yet I nevertheless love writing as a interest (and hate math, even inspite of the actuality that i'm no longer undesirable at it) so i will purely say English.

divide the equation first. then you get x+3 + 5/x+1, then take the integral.

I got x²/2 + 3x + 5ln |x+1| +c.

if u have any questions feel free to ask :)