What is the integral from x^2 to x of 4xy ?

The integral from x^2 to x of 4xy is 2x(-x^4+x^2)

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integral from x^2 to x of 4xy

\int_{x^{2}}^{x} 4 x y d y

2 x (- x^{4} + x^{2})

Solution steps

\int_{x^{2}}^{x} 4 x y d y

Take the constant out:

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

= 4 x \cdot \int_{x^{2}}^{x} y d y

Apply the Power Rule

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

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

Compute the boundaries:

\frac{x ^{2}}{2} - \frac{x ^{4}}{2}

= 4 x (\frac{x ^{2}}{2} - \frac{x ^{4}}{2})

Simplify

= 2 x (- x^{4} + x^{2})

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What is the integral from x^2 to x of 4xy ?
The integral from x^2 to x of 4xy is 2x(-x^4+x^2)