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

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

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

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

e^{x^{3}} + C

Solution steps

\int 3 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

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

Apply u-substitution

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

= 3 \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

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

Use the common integral:

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

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

Substitute back

u = x^{3}

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

Simplify

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

= e^{x^{3}}

Add a constant to the solution

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

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