What is the integral from-t to t^2 of e^{-x} ?

The integral from-t to t^2 of e^{-x} is -e^{-t^2}+e^t

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integral from-t to t^2 of e^{-x}

\int_{- t}^{t^{2}} e^{- x} d x

- e^{- t^{2}} + e^{t}

Solution steps

\int_{- t}^{t^{2}} e^{- x} d x

Apply u-substitution

: \int_{t}^{- t^{2}} - e^{u} d u

= \int_{t}^{- t^{2}} - e^{u} d u

Take the constant out:

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

= - \int_{t}^{- t^{2}} e^{u} d u

Use the common integral:

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

= - [e^{u}]_{t}^{- t^{2}}

Compute the boundaries:

e^{- t^{2}} - e^{t}

= - (e^{- t^{2}} - e^{t})

Simplify

= - e^{- t^{2}} + e^{t}

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What is the integral from-t to t^2 of e^{-x} ?
The integral from-t to t^2 of e^{-x} is -e^{-t^2}+e^t