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

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

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

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

- ln ∣ x - 1 ∣ + C

Solution steps

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

Simplify

\frac{1 - 2 x - x ^{2}}{x ^{3} + x ^{2} - 3 x + 1} : - \frac{1}{x - 1}

= \int - \frac{1}{x - 1} d x

Take the constant out:

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

= - \int \frac{1}{x - 1} d x

Apply u-substitution

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

= - \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

= - ln ∣ x - 1 ∣

Add a constant to the solution

= - ln ∣ x - 1 ∣ + C

x \to 3 lim (\frac{- x + 6}{- 2 - 1})

\frac{d}{d x} (k x e^{x})

x \to 2 lim (x^{2} + 5 x)

sin (x) e^{c o s (x)}

\int sin (7 x + 1) d x

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