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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int \frac{u ^{2} + 2}{u} d u

Expand

\frac{u ^{2} + 2}{u} : u + \frac{2}{u}

= \int u + \frac{2}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u d u + \int \frac{2}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{u ^{2}}{2} + 2 ln ∣ u ∣

Substitute back

u = x + 1

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

Add a constant to the solution

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

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