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

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

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

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

2 ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= 2 ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣ + C

- X^{3} + X, X = - 1, Y = - \frac{1}{2}

\frac{d}{d x} (arccos (\frac{x}{2}))

f (- 1) = 4 x^{2} + 4 x - 3

f (x) = (\frac{x}{x + 1})^{2}

\frac{\partial}{\partial y} (e^{x} (1 - cos (y)))

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