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

The integral of y/(x^2y^2+1) is arctan(yx)+C

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

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

arctan (y x) + C

Solution steps

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = y x

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

Simplify

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

= arctan (y x)

Add a constant to the solution

= arctan (y x) + C

9 - x^{2}

3 x - 5

\frac{d}{d x} ([x + (x + sin^{2} (x))^{5}]^{4})

y^{^{''}} - 4 y^{^{'}} + 20 y = 0

f (x) = (x - 1) (x + 1)^{3}

What is the integral of y/(x^2y^2+1) ?
The integral of y/(x^2y^2+1) is arctan(yx)+C