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

The integral of x/(4+3x) is 1/3 x-4/9 ln|3x+4|+C

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integral of x/(4+3x)

\int \frac{x}{4 + 3 x} d x

\frac{1}{3} x - \frac{4}{9} ln ∣ 3 x + 4 ∣ + C

Solution steps

\int \frac{x}{4 + 3 x} d x

Long division

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

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

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

\int \frac{4}{3 ( 3 x + 4 )} d x = \frac{4}{9} ln ∣ 3 x + 4 ∣

= \frac{1}{3} x - \frac{4}{9} ln ∣ 3 x + 4 ∣

Add a constant to the solution

= \frac{1}{3} x - \frac{4}{9} ln ∣ 3 x + 4 ∣ + C

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What is the integral of x/(4+3x) ?
The integral of x/(4+3x) is 1/3 x-4/9 ln|3x+4|+C