What is the integral from 0 to y of 2e^{-(x+y)} ?

The integral from 0 to y of 2e^{-(x+y)} is -2(e^{-2y}-e^{-y})

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integral from 0 to y of 2e^{-(x+y)}

\int_{0}^{y} 2 e^{- (x + y)} d x

- 2 (e^{- 2 y} - e^{- y})

Solution steps

\int_{0}^{y} 2 e^{- (x + y)} d x

Take the constant out:

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

= 2 \cdot \int_{0}^{y} e^{- (y + x)} d x

Apply u-substitution

: \int_{- y}^{- 2 y} - e^{u} d u

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

Take the constant out:

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

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

Use the common integral:

\int e^{u} d u = e^{u}

= 2 (- [e^{u}]_{- y}^{- 2 y})

Simplify

= - 2 [e^{u}]_{- y}^{- 2 y}

Compute the boundaries:

e^{- 2 y} - e^{- y}

= - 2 (e^{- 2 y} - e^{- y})

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What is the integral from 0 to y of 2e^{-(x+y)} ?
The integral from 0 to y of 2e^{-(x+y)} is -2(e^{-2y}-e^{-y})