What is the integral of 6t^2e^{-t^3} ?

The integral of 6t^2e^{-t^3} is -2e^{-t^3}+C

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integral of 6t^2e^{-t^3}

\int 6 t^{2} e^{- t^{3}} d t

- 2 e^{- t^{3}} + C

Solution steps

\int 6 t^{2} e^{- t^{3}} d t

Take the constant out:

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

= 6 \cdot \int t^{2} e^{- t^{3}} d t

Apply u-substitution

: \int - \frac{e ^{u}}{3} d u

= 6 \cdot \int - \frac{e ^{u}}{3} d u

Take the constant out:

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

= 6 (- \frac{1}{3} \cdot \int e^{u} d u)

Use the common integral:

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

= 6 (- \frac{1}{3} e^{u})

Substitute back

u = - t^{3}

= 6 (- \frac{1}{3} e^{- t^{3}})

Simplify

6 (- \frac{1}{3} e^{- t^{3}}) : - 2 e^{- t^{3}}

= - 2 e^{- t^{3}}

Add a constant to the solution

= - 2 e^{- t^{3}} + C

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What is the integral of 6t^2e^{-t^3} ?
The integral of 6t^2e^{-t^3} is -2e^{-t^3}+C