What is the integral from 0 to T of e^{-xy} ?

The integral from 0 to T of e^{-xy} is -(e^{-Ty}-1)/y

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integral from 0 to T of e^{-xy}

\int_{0}^{T} e^{- x y} d x

- \frac{e ^{- T y} - 1}{y}

Solution steps

\int_{0}^{T} e^{- x y} d x

Apply u-substitution

: \int_{0}^{- y T} - \frac{e ^{u}}{y} d u

= \int_{0}^{- y T} - \frac{e ^{u}}{y} d u

Take the constant out:

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

= - \frac{1}{y} \cdot \int_{0}^{- y T} e^{u} d u

Use the common integral:

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

= - \frac{1}{y} [e^{u}]_{0}^{- y T}

Simplify

- \frac{1}{y} [e^{u}]_{0}^{- y T} : - \frac{[ e ^{u} ] _{0}^{- T y}}{y}

= - \frac{[ e ^{u} ] _{0}^{- T y}}{y}

Compute the boundaries:

e^{- T y} - 1

= - \frac{e ^{- T y} - 1}{y}

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What is the integral from 0 to T of e^{-xy} ?
The integral from 0 to T of e^{-xy} is -(e^{-Ty}-1)/y