What is the integral of (2x^3+x^2-7x+7)/(x^2+x-2) ?

The integral of (2x^3+x^2-7x+7)/(x^2+x-2) is x^2-x+ln|x-1|-3ln|x+2|+C

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integral of (2x^3+x^2-7x+7)/(x^2+x-2)

\int \frac{2 x ^{3} + x ^{2} - 7 x + 7}{x ^{2} + x - 2} d x

x^{2} - x + ln ∣ x - 1 ∣ - 3 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{2 x ^{3} + x ^{2} - 7 x + 7}{x ^{2} + x - 2} d x

Take the partial fraction of

\frac{2 x ^{3} + x ^{2} - 7 x + 7}{x ^{2} + x - 2} : 2 x - 1 + \frac{1}{x - 1} - \frac{3}{x + 2}

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

Apply the Sum Rule:

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

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

\int 2 x d x = x^{2}

\int 1 d x = x

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

\int \frac{3}{x + 2} d x = 3 ln ∣ x + 2 ∣

= x^{2} - x + ln ∣ x - 1 ∣ - 3 ln ∣ x + 2 ∣

Add a constant to the solution

= x^{2} - x + ln ∣ x - 1 ∣ - 3 ln ∣ x + 2 ∣ + C

\frac{\partial}{\partial y} (x y e^{4 y})

\int_{1}^{4} 1 + \frac{9}{4} x d x

\frac{d}{d x} (\frac{x}{1 - 2 x ^{2}})

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

2 e^{- x}

What is the integral of (2x^3+x^2-7x+7)/(x^2+x-2) ?
The integral of (2x^3+x^2-7x+7)/(x^2+x-2) is x^2-x+ln|x-1|-3ln|x+2|+C