What is the integral of x/((x+4)(x-3)) ?

The integral of x/((x+4)(x-3)) is 4/7 ln|x+4|+3/7 ln|x-3|+C

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integral of x/((x+4)(x-3))

\int \frac{x}{( x + 4 ) ( x - 3 )} d x

\frac{4}{7} ln ∣ x + 4 ∣ + \frac{3}{7} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x}{( x + 4 ) ( x - 3 )} d x

Take the partial fraction of

\frac{x}{( x + 4 ) ( x - 3 )} : \frac{4}{7 ( x + 4 )} + \frac{3}{7 ( x - 3 )}

= \int \frac{4}{7 ( x + 4 )} + \frac{3}{7 ( 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

= \int \frac{4}{7 ( x + 4 )} d x + \int \frac{3}{7 ( x - 3 )} d x

\int \frac{4}{7 ( x + 4 )} d x = \frac{4}{7} ln ∣ x + 4 ∣

\int \frac{3}{7 ( x - 3 )} d x = \frac{3}{7} ln ∣ x - 3 ∣

= \frac{4}{7} ln ∣ x + 4 ∣ + \frac{3}{7} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{4}{7} ln ∣ x + 4 ∣ + \frac{3}{7} ln ∣ x - 3 ∣ + C

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What is the integral of x/((x+4)(x-3)) ?
The integral of x/((x+4)(x-3)) is 4/7 ln|x+4|+3/7 ln|x-3|+C