What is the integral from 0 to 2 of 6x^5e^{x^3y} ?

The integral from 0 to 2 of 6x^5e^{x^3y} is 6x^2(e^{2x^3}-1)

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integral from 0 to 2 of 6x^5e^{x^3y}

\int_{0}^{2} 6 x^{5} e^{x^{3} y} d y

6 x^{2} (e^{2 x^{3}} - 1)

Solution steps

\int_{0}^{2} 6 x^{5} e^{x^{3} y} d y

Take the constant out:

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

= 6 x^{5} \cdot \int_{0}^{2} e^{x^{3} y} d y

Apply u-substitution

: \int_{0}^{x^{3} \cdot 2} \frac{e ^{u}}{x ^{3}} d u

= 6 x^{5} \cdot \int_{0}^{x^{3} \cdot 2} \frac{e ^{u}}{x ^{3}} d u

Take the constant out:

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

= 6 x^{5} \frac{1}{x ^{3}} \cdot \int_{0}^{x^{3} \cdot 2} e^{u} d u

Use the common integral:

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

= 6 x^{5} \frac{1}{x ^{3}} [e^{u}]_{0}^{x^{3} \cdot 2}

Simplify

6 x^{5} \frac{1}{x ^{3}} [e^{u}]_{0}^{x^{3} \cdot 2} : 6 x^{2} [e^{u}]_{0}^{2 x^{3}}

= 6 x^{2} [e^{u}]_{0}^{2 x^{3}}

Compute the boundaries:

e^{2 x^{3}} - 1

= 6 x^{2} (e^{2 x^{3}} - 1)

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\int_{1}^{2} x^{5} e^{x^{2}} d x

What is the integral from 0 to 2 of 6x^5e^{x^3y} ?
The integral from 0 to 2 of 6x^5e^{x^3y} is 6x^2(e^{2x^3}-1)