What is the integral from l to u of r*e^{-t*r} ?

The integral from l to u of r*e^{-t*r} is -e^{-ur}+e^{-lr}

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integral from l to u of re^{-tr}

\int_{l}^{u} r \cdot e^{- t \cdot r} d t

- e^{- u r} + e^{- l r}

Solution steps

\int_{l}^{u} r e^{- t r} d t

Take the constant out:

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

= r \cdot \int_{l}^{u} e^{- r t} d t

Apply u-substitution

: \int_{- r l}^{- r u} - \frac{e ^{v}}{r} d v

= r \cdot \int_{- r l}^{- r u} - \frac{e ^{v}}{r} d v

Take the constant out:

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

= r (- \frac{1}{r} \cdot \int_{- r l}^{- r u} e^{v} d v)

Use the common integral:

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

= r (- \frac{1}{r} [e^{v}]_{- r l}^{- r u})

Simplify

r (- \frac{1}{r} [e^{v}]_{- r l}^{- r u}) : - [e^{v}]_{- l r}^{- u r}

= - [e^{v}]_{- l r}^{- u r}

Compute the boundaries:

e^{- u r} - e^{- l r}

= - (e^{- u r} - e^{- l r})

Simplify

= - e^{- u r} + e^{- l r}

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What is the integral from l to u of r*e^{-t*r} ?
The integral from l to u of r*e^{-t*r} is -e^{-ur}+e^{-lr}