What is the integral from x-1 to 1-x of 30xy^2 ?

The integral from x-1 to 1-x of 30xy^2 is 10x(-2x^3+6x^2-6x+2)

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integral from x-1 to 1-x of 30xy^2

\int_{x - 1}^{1 - x} 30 x y^{2} d y

10 x (- 2 x^{3} + 6 x^{2} - 6 x + 2)

Solution steps

\int_{x - 1}^{1 - x} 30 x y^{2} d y

Take the constant out:

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

= 30 x \cdot \int_{- 1 + x}^{1 - x} y^{2} d y

Apply the Power Rule

: [\frac{y ^{3}}{3}]_{- 1 + x}^{1 - x}

= 30 x [\frac{y ^{3}}{3}]_{- 1 + x}^{1 - x}

Compute the boundaries:

\frac{( 1 - x ) ^{3}}{3} - \frac{( - 1 + x ) ^{3}}{3}

= 30 x (\frac{( 1 - x ) ^{3}}{3} - \frac{( - 1 + x ) ^{3}}{3})

Simplify

= 10 x (- 2 x^{3} + 6 x^{2} - 6 x + 2)

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What is the integral from x-1 to 1-x of 30xy^2 ?
The integral from x-1 to 1-x of 30xy^2 is 10x(-2x^3+6x^2-6x+2)