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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Use the common integral:

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

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

Substitute back

u = x^{3}

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

Simplify

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

= y e^{x^{3}}

Add a constant to the solution

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

What is the integral of 3x^2ye^{x^3} ?
The integral of 3x^2ye^{x^3} is ye^{x^3}+C