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

The integral of (x^2-3x+2)/(x+1) is ((x+1)^2)/2-5(x+1)+6ln|x+1|+C

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

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

\frac{( x + 1 ) ^{2}}{2} - 5 (x + 1) + 6 ln ∣ x + 1 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{2} - 5 u + 6}{u} d u

= \int \frac{u ^{2} - 5 u + 6}{u} d u

Expand

\frac{u ^{2} - 5 u + 6}{u} : u - 5 + \frac{6}{u}

= \int u - 5 + \frac{6}{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 5 d u + \int \frac{6}{u} d u

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

\int 5 d u = 5 u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

= \frac{u ^{2}}{2} - 5 u + 6 ln ∣ u ∣

Substitute back

u = x + 1

= \frac{( x + 1 ) ^{2}}{2} - 5 (x + 1) + 6 ln ∣ x + 1 ∣

Add a constant to the solution

= \frac{( x + 1 ) ^{2}}{2} - 5 (x + 1) + 6 ln ∣ x + 1 ∣ + C

\int \frac{5 y ^{4}}{y ^{5} - 13} d y

\frac{50 s}{s ^{2} + 25}

\frac{\partial}{\partial x} (e^{- x} y)

\int_{- 2}^{5} - x^{3} + 9 x d x

\frac{d}{d x} (4 cot (x))

What is the integral of (x^2-3x+2)/(x+1) ?
The integral of (x^2-3x+2)/(x+1) is ((x+1)^2)/2-5(x+1)+6ln|x+1|+C