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

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

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

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

e^{- t + x} + C

Solution steps

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

Simplify

= \int e^{- t + x} d x

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

= \int e^{- t} e^{x} d x

Take the constant out:

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

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

Use the common integral:

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

= e^{- t} e^{x}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= e^{- t + x}

Add a constant to the solution

= e^{- t + x} + C