What is the integral of e^{x^3}(12x^2) ?

The integral of e^{x^3}(12x^2) is 4e^{x^3}+C

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integral of e^{x^3}(12x^2)

\int e^{x^{3}} (12 x^{2}) d x

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

Solution steps

\int e^{x^{3}} \cdot 12 x^{2} d x

Take the constant out:

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

= 12 \cdot \int e^{x^{3}} x^{2} d x

Apply u-substitution

= 12 \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

= 12 \cdot \frac{1}{3} \cdot \int e^{u} d u

Use the common integral:

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

= 12 \cdot \frac{1}{3} e^{u}

Substitute back

u = x^{3}

= 12 \cdot \frac{1}{3} e^{x^{3}}

Simplify

12 \cdot \frac{1}{3} e^{x^{3}} : 4 e^{x^{3}}

= 4 e^{x^{3}}

Add a constant to the solution

= 4 e^{x^{3}} + C

What is the integral of e^{x^3}(12x^2) ?
The integral of e^{x^3}(12x^2) is 4e^{x^3}+C