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

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

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

integral

\frac{1}{( x ( x + 2 ) )}

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

Add a constant to the solution

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

5^{- x}

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

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

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

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

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