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

The integral of (7x-2)/((x-2)(x+1)) is 4ln|x-2|+3ln|x+1|+C

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

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

4 ln ∣ x - 2 ∣ + 3 ln ∣ x + 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{7 x - 2}{( x - 2 ) ( x + 1 )} : \frac{4}{x - 2} + \frac{3}{x + 1}

= \int \frac{4}{x - 2} + \frac{3}{x + 1} 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 \frac{4}{x - 2} d x + \int \frac{3}{x + 1} d x

\int \frac{4}{x - 2} d x = 4 ln ∣ x - 2 ∣

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

= 4 ln ∣ x - 2 ∣ + 3 ln ∣ x + 1 ∣

Add a constant to the solution

= 4 ln ∣ x - 2 ∣ + 3 ln ∣ x + 1 ∣ + C

\frac{x}{x + 1}

\int \frac{x + 2}{x ^{2} + 4 x} d x

y = 4^{- 5 x + 2}

\frac{d}{d x} (4 sin (x^{2}))

x y^{^{'}} - 2 y = - 2 x^{2} + x + 1

What is the integral of (7x-2)/((x-2)(x+1)) ?
The integral of (7x-2)/((x-2)(x+1)) is 4ln|x-2|+3ln|x+1|+C