What is the integral of e^{-x^4}(-4x^3) ?

The integral of e^{-x^4}(-4x^3) is e^{-x^4}+C

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integral of e^{-x^4}(-4x^3)

\int e^{- x^{4}} (- 4 x^{3}) d x

e^{- x^{4}} + C

Solution steps

\int e^{- x^{4}} (- 4 x^{3}) d x

Simplify

= \int - 4 e^{- x^{4}} x^{3} d x

Take the constant out:

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

= - 4 \cdot \int e^{- x^{4}} x^{3} d x

Apply u-substitution

= - 4 \cdot \int - \frac{e ^{u}}{4} d u

Take the constant out:

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

= - 4 (- \frac{1}{4} \cdot \int e^{u} d u)

Use the common integral:

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

= - 4 (- \frac{1}{4} e^{u})

Substitute back

u = - x^{4}

= - 4 (- \frac{1}{4} e^{- x^{4}})

Simplify

- 4 (- \frac{1}{4} e^{- x^{4}}) : e^{- x^{4}}

= e^{- x^{4}}

Add a constant to the solution

= e^{- x^{4}} + C

What is the integral of e^{-x^4}(-4x^3) ?
The integral of e^{-x^4}(-4x^3) is e^{-x^4}+C