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

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

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

integral

\frac{( x + 2 )}{( x ^{2} + 3 x )}

\frac{2}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x + 3 ∣ + C

Solution steps

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

Expand

\frac{x + 2}{x ^{2} + 3 x} : \frac{x}{x ^{2} + 3 x} + \frac{2}{x ^{2} + 3 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{x}{x ^{2} + 3 x} d x + \int \frac{2}{x ^{2} + 3 x} d x

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

\int \frac{2}{x ^{2} + 3 x} d x = 2 (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 3 ∣)

= ln ∣ x + 3 ∣ + 2 (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 3 ∣)

Simplify

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

= \frac{2}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{2}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x + 3 ∣ + C

\frac{v}{2}

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

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

- (\frac{x}{( 1 - x )})

2^{10 x - 1}

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