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

The integral of (2ay)/(x^2+y^2) is 2aarctan(x/y)+C

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integral of (2ay)/(x^2+y^2)

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

2 a arctan (\frac{x}{y}) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 a y \cdot \int \frac{1}{y ^{2} + x ^{2}} d x

Apply Integral Substitution

= 2 a y \cdot \int \frac{1}{y ( u ^{2} + 1 )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 a y \frac{1}{y} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= 2 a y \frac{1}{y} arctan (u)

Substitute back

u = \frac{x}{y}

= 2 a y \frac{1}{y} arctan (\frac{x}{y})

Simplify

2 a y \frac{1}{y} arctan (\frac{x}{y}) : 2 a arctan (\frac{x}{y})

= 2 a arctan (\frac{x}{y})

Add a constant to the solution

= 2 a arctan (\frac{x}{y}) + C

x \to 1 lim (x^{2} + 2 x + 3)

cos (\frac{x}{x + 2})

\int \frac{1}{x 1 + x ^{4}} d x

f (x) = \frac{1}{( 2 x + 1 ) ^{2}}

\frac{d y}{d x} = \frac{7 x + 2 y}{7 x + 2 y + 2}

What is the integral of (2ay)/(x^2+y^2) ?
The integral of (2ay)/(x^2+y^2) is 2aarctan(x/y)+C