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

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

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

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

\frac{8}{3} e^{x^{3}} + C

Solution steps

\int 2 \cdot 4 x^{2} e^{x^{3}} d x

Take the constant out:

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

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

Simplify

= 8 \cdot \int x^{2} e^{x^{3}} d x

Apply u-substitution

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

= 8 \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

= 8 \cdot \frac{1}{3} \cdot \int e^{u} d u

Use the common integral:

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

= 8 \cdot \frac{1}{3} e^{u}

Substitute back

u = x^{3}

= 8 \cdot \frac{1}{3} e^{x^{3}}

Simplify

8 \cdot \frac{1}{3} e^{x^{3}} : \frac{8}{3} e^{x^{3}}

= \frac{8}{3} e^{x^{3}}

Add a constant to the solution

= \frac{8}{3} e^{x^{3}} + C

x \to 0 + lim (x + ln (x))

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