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

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

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integral of e^{x^3}6x^2

\int e^{x^{3}} 6 x^{2} d x

2 e^{x^{3}} + C

Solution steps

\int e^{x^{3}} \cdot 6 x^{2} d x

Take the constant out:

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

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

Apply u-substitution

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

Use the common integral:

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

= 6 \cdot \frac{1}{3} e^{u}

Substitute back

u = x^{3}

= 6 \cdot \frac{1}{3} e^{x^{3}}

Simplify

6 \cdot \frac{1}{3} e^{x^{3}} : 2 e^{x^{3}}

= 2 e^{x^{3}}

Add a constant to the solution

= 2 e^{x^{3}} + C