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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

= x \frac{1}{x} arctan (u)

Substitute back

u = \frac{y}{x}

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

Simplify

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

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

Add a constant to the solution

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

f (x) = 9 e^{x}, at x = 7

\int_{1}^{12} \frac{1}{x - 7} d x

\frac{\partial}{\partial y} (x^{2} + 6 x y + 4 y^{2})

\int x^{2} + 2 x + 1 d x

\frac{x ^{2} - 1}{2 x - 3}

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