What is the integral of e^{4x^2}(-4x^3) ?

The integral of e^{4x^2}(-4x^3) is -1/8 (4e^{4x^2}x^2-e^{4x^2})+C

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integral of e^{4x^2}(-4x^3)

\int e^{4 x^{2}} (- 4 x^{3}) d x

- \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}}) + C

Solution steps

\int e^{4 x^{2}} (- 4 x^{3}) d x

Simplify

= \int - 4 e^{4 x^{2}} 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^{4 x^{2}} x^{3} d x

Apply u-substitution

: \int \frac{e ^{u} u}{32} d u

= - 4 \cdot \int \frac{e ^{u} u}{32} d u

Take the constant out:

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

= - 4 \cdot \frac{1}{32} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= - 4 \cdot \frac{1}{32} (e^{u} u - \int e^{u} d u)

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

= - 4 \cdot \frac{1}{32} (e^{u} u - e^{u})

Substitute back

u = 4 x^{2}

= - 4 \cdot \frac{1}{32} (e^{4 x^{2}} \cdot 4 x^{2} - e^{4 x^{2}})

Simplify

- 4 \cdot \frac{1}{32} (e^{4 x^{2}} \cdot 4 x^{2} - e^{4 x^{2}}) : - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}})

= - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}})

Add a constant to the solution

= - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}}) + C

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What is the integral of e^{4x^2}(-4x^3) ?
The integral of e^{4x^2}(-4x^3) is -1/8 (4e^{4x^2}x^2-e^{4x^2})+C