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

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

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

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

\frac{1}{3} ln (∣ x + 1 ∣ x^{2} + 2 x + 4) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = (x + 1) (x^{2} + 2 x + 4)

= \frac{1}{3} ln (x + 1) (x^{2} + 2 x + 4)

Simplify

\frac{1}{3} ln (x + 1) (x^{2} + 2 x + 4) : \frac{1}{3} ln (∣ x + 1 ∣ x^{2} + 2 x + 4)

= \frac{1}{3} ln (∣ x + 1 ∣ x^{2} + 2 x + 4)

Add a constant to the solution

= \frac{1}{3} ln (∣ x + 1 ∣ x^{2} + 2 x + 4) + C

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