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

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

\int \frac{1}{12 x - 4 x ^{2} - 8} d x

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

\frac{d}{d x} (2 x^{7} - 7 x^{- 4} + 2 x^{2} - 3 x)

x \to \infty lim (\frac{x + 3}{2 x ^{3} - 5})

3^{x} ln (3)

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