What is the integral from 0 to 1-x of x^2y^2 ?

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

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integral from 0 to 1-x of x^2y^2

\int_{0}^{1 - x} x^{2} y^{2} d y

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

Solution steps

\int_{0}^{1 - x} x^{2} y^{2} d y

Take the constant out:

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

= x^{2} \cdot \int_{0}^{1 - x} y^{2} d y

Apply the Power Rule

: [\frac{y ^{3}}{3}]_{0}^{1 - x}

= x^{2} [\frac{y ^{3}}{3}]_{0}^{1 - x}

Compute the boundaries:

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

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

Simplify

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

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