What is the integral from 0 to 1 of y/(x^2y^2+1) ?

The integral from 0 to 1 of y/(x^2y^2+1) is arctan(y)

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integral from 0 to 1 of y/(x^2y^2+1)

\int_{0}^{1} \frac{y}{x ^{2} y ^{2} + 1} d x

arctan (y)

Solution steps

\int_{0}^{1} \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_{0}^{1} \frac{1}{1 + y ^{2} x ^{2}} d x

Apply Integral Substitution

= y \cdot \int_{0}^{y} \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_{0}^{y} \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)]_{0}^{y}

Simplify

y \frac{1}{y} [arctan (u)]_{0}^{y} : [arctan (u)]_{0}^{y}

= [arctan (u)]_{0}^{y}

Compute the boundaries:

arctan (y)

= arctan (y)

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What is the integral from 0 to 1 of y/(x^2y^2+1) ?
The integral from 0 to 1 of y/(x^2y^2+1) is arctan(y)