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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = \frac{x}{z}

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

Simplify

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

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

Add a constant to the solution

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

\int (sin (x))^{2} (cos (x)) d x

\frac{d}{d x} (6 x^{\frac{7}{2}} + 4 x^{\frac{5}{2}} + 2 x)

\frac{e ^{- π s}}{s - 1}

\frac{d}{d x} (2 x^{6})

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