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

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

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integral of x/(x+2y)

\int \frac{x}{x + 2 y} d x

\frac{x ^{2}}{x + 2 y} - 2 y (ln ∣ x + 2 y ∣ + \frac{2 y}{x + 2 y}) + C

Solution steps

\int \frac{x}{x + 2 y} d x

Apply Integration By Parts

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

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

= \frac{x ^{2}}{x + 2 y} - 2 y (ln ∣ x + 2 y ∣ + \frac{2 y}{x + 2 y})

Add a constant to the solution

= \frac{x ^{2}}{x + 2 y} - 2 y (ln ∣ x + 2 y ∣ + \frac{2 y}{x + 2 y}) + C

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