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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u ^{2} + 1}{u} d u

= \int \frac{u ^{2} + 1}{u} d u

Expand

\frac{u ^{2} + 1}{u} : u + \frac{1}{u}

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} + ln ∣ u ∣

Substitute back

u = x + 1

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

Add a constant to the solution

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

\frac{d}{d x} (sec^{2} (\frac{x}{a}) + csc^{2} (\frac{x}{a}))

f (x) = x^{2} - 6, (2, - 2)

x \to - \infty lim (2. 3^{x})

2 y^{^{''}} + 35 y = 0

x \to 2 lim (\frac{x ^{2}}{x ^{2} + 1})

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