What is the integral of e^{-x}f(e^{-x}) ?

The integral of e^{-x}f(e^{-x}) is -1/2 fe^{-2x}+C

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integral of e^{-x}f(e^{-x})

\int e^{- x} f (e^{- x}) d x

- \frac{1}{2} f e^{- 2 x} + C

Solution steps

\int e^{- x} f e^{- x} d x

Take the constant out:

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

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

Simplify

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

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

Apply u-substitution

= f \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

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

Use the common integral:

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

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

Substitute back

u = - 2 x

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

Simplify

= - \frac{1}{2} f e^{- 2 x}

Add a constant to the solution

= - \frac{1}{2} f e^{- 2 x} + C

What is the integral of e^{-x}f(e^{-x}) ?
The integral of e^{-x}f(e^{-x}) is -1/2 fe^{-2x}+C