What is the integral from 1-x to sqrt(x of)x^2y ?

The integral from 1-x to sqrt(x of)x^2y is (x^2(-x^2+3x-1))/2

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integral from 1-x to sqrt(x of)x^2y

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

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

Solution steps

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

Take the constant out:

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

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

Apply the Power Rule

: [\frac{y ^{2}}{2}]_{1 - x}^{x}

= x^{2} [\frac{y ^{2}}{2}]_{1 - x}^{x}

Compute the boundaries:

\frac{x}{2} - \frac{( 1 - x ) ^{2}}{2}

= x^{2} (\frac{x}{2} - \frac{( 1 - x ) ^{2}}{2})

Simplify

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

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What is the integral from 1-x to sqrt(x of)x^2y ?
The integral from 1-x to sqrt(x of)x^2y is (x^2(-x^2+3x-1))/2