What is the integral of 3/([x(x+3)]) ?

The integral of 3/([x(x+3)]) is ln|x|-ln|x+3|+C

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integral of 3/([x(x+3)])

\int \frac{3}{[ x ( x + 3 ) ]} d x

ln ∣ x ∣ - ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{3}{[ x ( x + 3 ) ]} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{1}{[ x ( x + 3 ) ]} d x

Simplify

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

Take the partial fraction of

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

= 3 \cdot \int \frac{1}{3 x} - \frac{1}{3 ( x + 3 )} 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

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

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

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

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

Simplify

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

= ln ∣ x ∣ - ln ∣ x + 3 ∣

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ x + 3 ∣ + C

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