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

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

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

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

- e^{- 2 x} + e^{- x}

Solution steps

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

Apply u-substitution

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

= \int_{- x}^{- 2 x} - e^{u} d u

Take the constant out:

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

= - \int_{- x}^{- 2 x} e^{u} d u

Use the common integral:

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

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

Compute the boundaries:

e^{- 2 x} - e^{- x}

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

Simplify

= - e^{- 2 x} + e^{- x}

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