What is the integral of x/((x+1)(x+3)(x+5)) ?

The integral of x/((x+1)(x+3)(x+5)) is -1/8 ln|x+1|+3/4 ln|x+3|-5/8 ln|x+5|+C

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integral of x/((x+1)(x+3)(x+5))

\int \frac{x}{( x + 1 ) ( x + 3 ) ( x + 5 )} d x

- \frac{1}{8} ln ∣ x + 1 ∣ + \frac{3}{4} ln ∣ x + 3 ∣ - \frac{5}{8} ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x}{( x + 1 ) ( x + 3 ) ( x + 5 )} d x

Take the partial fraction of

\frac{x}{( x + 1 ) ( x + 3 ) ( x + 5 )} : - \frac{1}{8 ( x + 1 )} + \frac{3}{4 ( x + 3 )} - \frac{5}{8 ( x + 5 )}

= \int - \frac{1}{8 ( x + 1 )} + \frac{3}{4 ( x + 3 )} - \frac{5}{8 ( x + 5 )} 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}{8 ( x + 1 )} d x + \int \frac{3}{4 ( x + 3 )} d x - \int \frac{5}{8 ( x + 5 )} d x

\int \frac{1}{8 ( x + 1 )} d x = \frac{1}{8} ln ∣ x + 1 ∣

\int \frac{3}{4 ( x + 3 )} d x = \frac{3}{4} ln ∣ x + 3 ∣

\int \frac{5}{8 ( x + 5 )} d x = \frac{5}{8} ln ∣ x + 5 ∣

= - \frac{1}{8} ln ∣ x + 1 ∣ + \frac{3}{4} ln ∣ x + 3 ∣ - \frac{5}{8} ln ∣ x + 5 ∣

Add a constant to the solution

= - \frac{1}{8} ln ∣ x + 1 ∣ + \frac{3}{4} ln ∣ x + 3 ∣ - \frac{5}{8} ln ∣ x + 5 ∣ + C

\int \frac{2}{( x - 7 ) ( x - 5 )} d x

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What is the integral of x/((x+1)(x+3)(x+5)) ?
The integral of x/((x+1)(x+3)(x+5)) is -1/8 ln|x+1|+3/4 ln|x+3|-5/8 ln|x+5|+C