What is the integral of 2e^{-x-2y} ?

The integral of 2e^{-x-2y} is -e^{-x-2y}+C

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integral of 2e^{-x-2y}

\int 2 e^{- x - 2 y} d y

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

Solution steps

\int 2 e^{- x - 2 y} d y

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int - \frac{1}{2} e^{u} d u

Take the constant out:

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

= 2 (- \frac{1}{2} \cdot \int e^{u} d u)

Use the common integral:

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

= 2 (- \frac{1}{2} e^{u})

Substitute back

u = - x - 2 y

= 2 (- \frac{1}{2} e^{- x - 2 y})

Simplify

2 (- \frac{1}{2} e^{- x - 2 y}) : - e^{- x - 2 y}

= - e^{- x - 2 y}

Add a constant to the solution

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