What is the integral of x^2e^{-2x^3+5} ?

The integral of x^2e^{-2x^3+5} is -1/6 e^{-2x^3+5}+C

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integral of x^2e^{-2x^3+5}

\int x^{2} e^{- 2 x^{3} + 5} d x

- \frac{1}{6} e^{- 2 x^{3} + 5} + C

Solution steps

\int x^{2} e^{- 2 x^{3} + 5} d x

Apply u-substitution

: \int \frac{e ^{- 2 u^{\frac{3}{2}} + 5} u}{2 x} d u

= \int \frac{e ^{- 2 u + 5}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int e^{- 2 u + 5} d u

Apply u-substitution

: \int - \frac{1}{2} e^{v} d v

= \frac{1}{3} \cdot \int - \frac{1}{2} e^{v} d v

Take the constant out:

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

= \frac{1}{3} (- \frac{1}{2} \cdot \int e^{v} d v)

Use the common integral:

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

= \frac{1}{3} (- \frac{1}{2} e^{v})

Substitute back

= \frac{1}{3} (- \frac{1}{2} e^{- 2 x^{3} + 5})

Simplify

\frac{1}{3} (- \frac{1}{2} e^{- 2 x^{3} + 5}) : - \frac{1}{6} e^{- 2 x^{3} + 5}

= - \frac{1}{6} e^{- 2 x^{3} + 5}

Add a constant to the solution

= - \frac{1}{6} e^{- 2 x^{3} + 5} + C

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What is the integral of x^2e^{-2x^3+5} ?
The integral of x^2e^{-2x^3+5} is -1/6 e^{-2x^3+5}+C