What is the integral of (x^2-7)/((x-2)(x+1)) ?

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

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integral of (x^2-7)/((x-2)(x+1))

\int \frac{x ^{2} - 7}{( x - 2 ) ( x + 1 )} d x

x + 2 ln ∣ x + 1 ∣ - ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x ^{2} - 7}{( x - 2 ) ( x + 1 )} d x

Long division

\frac{x ^{2} - 7}{( x - 2 ) ( x + 1 )} : 1 + \frac{x - 5}{x ^{2} - x - 2}

= \int 1 + \frac{x - 5}{x ^{2} - 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 1 d x + \int \frac{x - 5}{x ^{2} - x - 2} d x

\int 1 d x = x

\int \frac{x - 5}{x ^{2} - x - 2} d x = 2 ln ∣ x + 1 ∣ - ln ∣ x - 2 ∣

= x + 2 ln ∣ x + 1 ∣ - ln ∣ x - 2 ∣

Add a constant to the solution

= x + 2 ln ∣ x + 1 ∣ - ln ∣ x - 2 ∣ + C

\int \frac{cot ^{2} ( x )}{sin ^{2} ( x )} d x

cos (π t)

\int_{1}^{3} - x^{2} + 4 x - 3 d x

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

\frac{d}{d x} (1 - \frac{x}{2} + \frac{x ^{2}}{9})

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