What is the integral from 0 to x of (xy)/(1+x^4) ?

The integral from 0 to x of (xy)/(1+x^4) is (x^3)/(2(1+x^4))

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integral from 0 to x of (xy)/(1+x^4)

\int_{0}^{x} \frac{x y}{1 + x ^{4}} d y

\frac{x ^{3}}{2 ( 1 + x ^{4} )}

Solution steps

\int_{0}^{x} \frac{x y}{1 + x ^{4}} d y

Take the constant out:

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

= \frac{x}{1 + x ^{4}} \cdot \int_{0}^{x} y d y

Apply the Power Rule

: [\frac{y ^{2}}{2}]_{0}^{x}

= \frac{x}{1 + x ^{4}} [\frac{y ^{2}}{2}]_{0}^{x}

Compute the boundaries:

\frac{x ^{2}}{2}

= \frac{x}{1 + x ^{4}} \cdot \frac{x ^{2}}{2}

Simplify

= \frac{x ^{3}}{2 ( 1 + x ^{4} )}

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What is the integral from 0 to x of (xy)/(1+x^4) ?
The integral from 0 to x of (xy)/(1+x^4) is (x^3)/(2(1+x^4))