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

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

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

\int \frac{2 y + x}{1 + x} d x

1 + x - ln ∣ 1 + x ∣ + 2 y ln ∣ 1 + x ∣ + C

Solution steps

\int \frac{2 y + x}{1 + x} d x

Expand

\frac{x + 2 y}{1 + x} : \frac{x}{1 + x} + \frac{2 y}{1 + 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{x}{1 + x} d x + \int \frac{2 y}{1 + x} d x

\int \frac{x}{1 + x} d x = 1 + x - ln ∣ 1 + x ∣

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

= 1 + x - ln ∣ 1 + x ∣ + 2 y ln ∣ 1 + x ∣

Add a constant to the solution

= 1 + x - ln ∣ 1 + x ∣ + 2 y ln ∣ 1 + x ∣ + C

y = (14 x - x^{2})^{3}

n = 1 \sum \infty n^{15}

x (x^{3} - 1)^{6}

\int \frac{1}{x ^{2} + 8 x + 97} d x

\int e^{7 x} cos (5 x) d x

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