What is the integral of ye^{-x} ?

The integral of ye^{-x} is -ye^{-x}+C

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integral of ye^{-x}

\int y e^{- x} d x

- y e^{- x} + C

Solution steps

\int y e^{- x} d x

Take the constant out:

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

= y \cdot \int e^{- x} d x

Apply u-substitution

: \int - e^{u} d u

= y \cdot \int - e^{u} d u

Take the constant out:

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

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

Use the common integral:

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

= y (- e^{u})

Substitute back

u = - x

= y (- e^{- x})

Simplify

= - y e^{- x}

Add a constant to the solution

= - y e^{- x} + C

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