What is the integral of (3x^2+4x+2)/((x+1)(x^2+x+1)) ?

The integral of (3x^2+4x+2)/((x+1)(x^2+x+1)) is ln(|x+1||x^2+x+1|)+C

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integral of (3x^2+4x+2)/((x+1)(x^2+x+1))

\int \frac{3 x ^{2} + 4 x + 2}{( x + 1 ) ( x ^{2} + x + 1 )} d x

ln (∣ x + 1 ∣ x^{2} + x + 1) + C

Solution steps

\int \frac{3 x ^{2} + 4 x + 2}{( x + 1 ) ( x ^{2} + x + 1 )} d x

Apply u-substitution

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = (x + 1) (x^{2} + x + 1)

= ln (x + 1) (x^{2} + x + 1)

Simplify

ln (x + 1) (x^{2} + x + 1) : ln (∣ x + 1 ∣ x^{2} + x + 1)

= ln (∣ x + 1 ∣ x^{2} + x + 1)

Add a constant to the solution

= ln (∣ x + 1 ∣ x^{2} + x + 1) + C

What is the integral of (3x^2+4x+2)/((x+1)(x^2+x+1)) ?
The integral of (3x^2+4x+2)/((x+1)(x^2+x+1)) is ln(|x+1||x^2+x+1|)+C