What is the integral of 25t^4e^{-t^5} ?

The integral of 25t^4e^{-t^5} is -5e^{-t^5}+C

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integral of 25t^4e^{-t^5}

\int 25 t^{4} e^{- t^{5}} d t

- 5 e^{- t^{5}} + C

Solution steps

\int 25 t^{4} e^{- t^{5}} d t

Take the constant out:

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

= 25 \cdot \int t^{4} e^{- t^{5}} d t

Apply u-substitution

: \int - \frac{e ^{u}}{5} d u

= 25 \cdot \int - \frac{e ^{u}}{5} d u

Take the constant out:

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

= 25 (- \frac{1}{5} \cdot \int e^{u} d u)

Use the common integral:

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

= 25 (- \frac{1}{5} e^{u})

Substitute back

u = - t^{5}

= 25 (- \frac{1}{5} e^{- t^{5}})

Simplify

25 (- \frac{1}{5} e^{- t^{5}}) : - 5 e^{- t^{5}}

= - 5 e^{- t^{5}}

Add a constant to the solution

= - 5 e^{- t^{5}} + C

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What is the integral of 25t^4e^{-t^5} ?
The integral of 25t^4e^{-t^5} is -5e^{-t^5}+C