What is the integral of 20e^{-20(x-12.5)} ?

The integral of 20e^{-20(x-12.5)} is -e^{-20(x-12.5)}+C

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integral of 20e^{-20(x-12.5)}

\int 20 e^{- 20 (x - 12.5)} d x

- e^{- 20 (x - 12.5)} + C

Solution steps

\int 20 e^{- 20 (x - 12.5)} d x

Take the constant out:

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

= 20 \cdot \int e^{- 20 (x - 12.5)} d x

Apply u-substitution

= 20 \cdot \int - \frac{1}{20} e^{u} d u

Take the constant out:

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

= 20 (- \frac{1}{20} \cdot \int e^{u} d u)

Use the common integral:

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

= 20 (- \frac{1}{20} e^{u})

Substitute back

u = - 20 (x - 12.5)

= 20 (- \frac{1}{20} e^{- 20 (x - 12.5)})

Simplify

20 (- \frac{1}{20} e^{- 20 (x - 12.5)}) : - e^{- 20 (x - 12.5)}

= - e^{- 20 (x - 12.5)}

Add a constant to the solution

= - e^{- 20 (x - 12.5)} + C

What is the integral of 20e^{-20(x-12.5)} ?
The integral of 20e^{-20(x-12.5)} is -e^{-20(x-12.5)}+C