What is the integral from-y to y of c(y^2-x^2)e^{-y} ?

The integral from-y to y of c(y^2-x^2)e^{-y} is ce^{-y}(2y^3-(2y^3)/3)

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integral from-y to y of c(y^2-x^2)e^{-y}

\int_{- y}^{y} c (y^{2} - x^{2}) e^{- y} d x

c e^{- y} (2 y^{3} - \frac{2 y ^{3}}{3})

Solution steps

\int_{- y}^{y} c (y^{2} - x^{2}) e^{- y} d x

Take the constant out:

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= c e^{- y} (\int_{- y}^{y} y^{2} d x - \int_{- y}^{y} x^{2} d x)

\int_{- y}^{y} y^{2} d x = 2 y^{3}

\int_{- y}^{y} x^{2} d x = \frac{2 y ^{3}}{3}

= c e^{- y} (2 y^{3} - \frac{2 y ^{3}}{3})

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What is the integral from-y to y of c(y^2-x^2)e^{-y} ?
The integral from-y to y of c(y^2-x^2)e^{-y} is ce^{-y}(2y^3-(2y^3)/3)